Article 2 is The four Hagge Circles . This describes properties of the four Hagge circles of triangles BCD, ACD, ABD, ABC when ABCD is a cyclic quadrilateral.
Article 3 is Generalizations of Hagge's Theorem . Two possible generalizations of Hagge circles are considered. The first describes those properties preserved when the circle passes through a point other than the orthocentre. The second when a pair of orthologic triangles are involved.
Article 4 is Studies in Similarity . The six points: the orthocentre, the Brocard points and the points where the medians intersect the orthocentroidal circle involve nine circles through the vertices of ABC and either the orthocentre or one of the Brocard points. The triangles formed by the centres of these circles exhibit many similarities.
Article 5 is Omega circles . It is shown how circles through either of the Brocard points have properties similar to Hagge circles.
Article 6 is On the Ten point Wood-Desargues' Configuration . The ten pairs of directly similar triangles in perspective in the Wood-Desargues' configuration have ten common Hagge circle centres. It is shown in this paper that these centres lie at the vertices of five cyclic quadrangles, which are similar to the five cyclic quadrangles of the initial Wood-Desargues' configuration.
Article 7 is Generalization of the Wallace-Simson line. In this article it is shown how a direct similarity transformation of triangle ABC into another triangle enables one to identify the Double Wallace-Simson line of the new triangle as a generalized Wallace-Simson line of ABC.
Article 8 is A singular Miquel configuration and the Miquel direct similarity. In the Miquel configuration for a triangle the centres of the three concurrent circles form a triangle similar to ABC. It is shown how the direct similarity between the two triangles involved produces further results of significance.
Article 9 is Miquel circles and Cevian lines. The Miquel configuration has interesting additional properties when the points on the sides of ABC are the feet of Cevian lines.
Article 10 is A new construction to find any circle through a given point. This is another generalization of the Hagge construction when the point involved is not the orthocentre.
Article 11 is Eight Circles through the Orthocentre.
Article 12 is Circles concentric with the Circumcircle. It is shown that such circles contain seven points with special properties.
Article 13 is Ex-points and Eight Intersecting Circles. Points internal to a triangle have three points external to the triangle associated with them. Properties of these Ex-points are investigated.
Article 14 is Intersecting Circles having Chords the sides of a Cyclic Quadrilateral. The circles pass through the centre of the cyclic quadrilateral or the intersection of its diagonals.
Article 15 is Four concurrent Euler Lines. In this paper we consider a cyclic quadrilateral PQRS in which the diagonals PR and QS meet at a point E and we prove a number of results about the triangles PQE, QRE, RSE, SPE.
Article 16 is The Direct Similarity of the Miquel Point Configuration . This continues work begun in Article 8 and is best read in conjunction with it.
Article 17 is Harmonic Ranges in a Coaxal system of Circles. A circle cuts a triangle in six points and the condition that the harmonic conjugates of those points also lie on a circle is investigated. The coaxal system involving the polar circle, the circumcircle and the orthocentroidal circle feature.
Article 18 is Some Special Circles in a Triangle. There is one special circle through each point in the plane of ABC, not on the sides or the circumcircle. Eight points on each circle have similar properties to those in Hagge circles.
Article 19 is Circular Perspective. Two triangles ABC and UVW are in circular perspective when circles AVW, BWU, CUV all pass through the same point. It is shown that this concept is symmetric and two triangles in double circular perspective are automatically in triple circular perspective.
Article 20 is On the Nine Intersections of two Cevian Triangles. A Cevian triangle is one whose sides join the three feet of a Cevian point. Two Cevian triangles intersect in nine points which exhibit a vast number of properties.
Article 21 is Significant Points on Circles Centre the Circumcentre. Given a triangle ABC with circumcentre O and a point P not on its sides or their extensions and not on the circumcircle, it is shown that one may construct on the circle centre O and radius OP six significant points.
Article 22 is Six Points on a Circle. A construction with a triangle ABC and the point X76 is described, yielding a circle bearing a distinct similarity to the Triplicate Ratio Circle.
Article 23 is The Symmedian point and the Polar Line. A construction based on the Symmedian point is described yielding six lines through each of three points on the polar line. Two related porisms are constructed.
Article 24 is The Thirteen Point Circle. In this article we give an account of the properties of the coaxal system of circles passing through the two Brocard points and having the Brocard axis as line of centres.
Article 25 is When Quadrangles are completely in Perspective . In this article we establish a sufficient condition for when a pair of quadrangles have what may be appropriately called a Desargues' axis of perspective.
Article 26 is A Circle concentric with the Incircle. A construction is given in a triangle using Gergonne's point yielding a circle concentric with the incircle.
Article 27 is Porisms with a circular circumconic. The result we prove in this article is if we take the orthocentre of all the triangles in the porism in a case when the circumconic is the circumcircle, then the path traced out by the orthocentre is circular (or linear).
Article 28 is Some circles in a Cyclic Quadrilateral. The properties of some circles in a cyclic quadrilateral passing through its vertices, midpoints of sides and centre are investigated.
Article 29 is The Miquel Circles for a Quadrilateral. It is shown how these may be constructed starting from the Miquel points of triangles involving the diagonal points.
Article 30 is More on Circular Perspective. It is shown how a triangle may be in circular perspective with three points on a line (rather than a second triangle). An example involving the intersections of the tangents at the vertices of ABC with the opposite sides is given.
Article 31 is More cases of Circular Perspective . This article involves triangles ABC and PQR, where P, Q, R are the intersections of the medians with the orthocentroidal circle.
Article 32 is The GH Disc and anogther case of Triple Circular Perspective, and continues the work on triple circular perspective, this time in connection with the orthocentroidal circle.
Article 33 is 29 Circles, and continues the work on exploring triple circular perspective.
Article 34 is Some circles in the Cyclic Quadrilateral configuration. When ABCD is a cyclic quadrilateral, centre O, and BA^CD = E, BC^AD = F and AC^BD = G, then circles ABF, CDF, BCE, ADE meet at a point T on EF. The perpendicular from T to EF passes through G and O. The centres of the four circles, O and T lie on a circle.
Article 35 is Where 7 circles meet Part 1. The orthocentroidal circle S of triangle ABC on GH as diameter possesses an unusual property. If you draw circles BHC, CHA, AHB to meet S again at points X, Y, Z and circles BGC, CGA, AGB to meet S again at U, V, W then the following property holds: circles AYX, BZX, CXY, AVW, BWU, CUV all pass through a point Q on the circumcircle of ABC.
Article 36 is Where 7 circles meet Part 1. The result of Article 35 is generalized to involve an arbitrary circle.
Article 37 is More Circles in the Cyclic Quadrilateral Configuration. Quadrilateral ABCD, circle centre O has external diagonal points E and F. Circles on AO, CO as diameters meet at Y and circles on BO, DO as diameters meet at Z. It is shown that T, Y, Z are collinear, where T is the midpoint of EF.
Article 38 is Eight Points on a line and Seven Circles through a Point. Points D, X, Y, Z and E, U, V, W lie on a transversal of triangle ABC. It is shown what the condition is connecting D and E for circles ABC, AYZ, BZX, CXY, AVW, BWU, CUV to meet at a point.
Article 39 is The 60°, 75°, 45° triangle and its Euler line. Key results concerning the intersection of the Euler line and the sides of the triangle are established.
Article 40 is The 120°, 45°, 15° triangle and its Euler line. Many results involving the triangle, its Euler line and various circles are established.
Article 41 is The 105°, 60°, 15° Triangle and its Euler Line and Circles. Some remarkable properties of the 105°, 60°, 15° triangle and its intersections with its Euler line are investigated.
Article 42 is A Cyclic Quadrilateral, 29 Points and 33 Lines. It is shown that the quadrilateral formed by the symmedian points of triangle BDC, ACD, ABD, ABC of a cyclic quadrilateral ABCD has the same diagonal point triangle as the cyclic quadrilateral ABCD.
Article 43 is The Altitudes and Radii of a triangle and its circumcircle. In triangle ABC, circumcentre O and orthocentre H if ab is the intersection of AH and OB, then the midpoints of pairs ab, ac and ba, bc and ca, cb lie on the circle on OH as diameter and circles ab bc ca and ba ac cb both pass through H.
Article 44 is A natural Enlargement resulting in a Collineation . This is a short survey of the properties of the Exsimilicentre X56 and shows that it is on the line joining Feuerbach point and the Orthocentre. This is a known collineation.
Article 45 is On the Circles defining the Brocard points . A construction is given in which points X, Y, Z lie on a circle containing the first Brocard point and points X', Y', Z' lie on a circle containing the second Brocard point. The six circles such as BXC are the Brocard circles and the six circles such as AYZ always pass through Tarry's oint.
Article 46 is A Mean of two Cevian points and the Construction of Triangle centres . A construction is given which gives rise to about 9 million more triangle centres. It is part serious and part jocular, but its aim is to question the activity of hunting triangle centres.
Article 47 is A Cevian point and its six harmonics If L is the foot of a Cevian point on BC then B1 is the harmonic conjugate of L and C and C1 is the harmonic conjugate of L and B. When similar points on CA and AB are defined a configuration is produced with many interesting features.
Article 48 is The Symmedian point and its Harmonics Results of Article 47 are applied when the Cevian point is the Symmedian point.
Article 49 is Perpendicular Bisectors and Angle Bisectors When these are drawn for a given triangle the six non-trivial points of intersection create a figure with many significant results
Article 50 is Three triangles in mutual triple perspective . A configuration with pairs of points that are isotomic conjugates produces a figure with three triangles in mutual triple perspective. Six of the perspectrices pass through a given point.
Article 51 is When one Conic produces two more . ABC is a triangle, circumcentre O. The given conic is ABCLMN, where L, M, N lie on AO, BO, CO. Twelve points are then formed from these points which are shown to lie 6 by 6 on two conics.
Article 52 is Three triangles in mutual Triple Reverse Perspective . A construction and supporting analysis is given that ensures the perspectives. When two of the triangles have vertices on the circumcircle the Brocard porism is established and the third triangle degenerates and becomes the polar of the Symmedian point with respect to the circumcircle.
Article 53 is Hagge Circles touching at H . A construction of two triangles is described whose circumcircles turn out to be a pair of touching Hagge circles. One of the triangles has a vertex at H and an interesting connection with the nine-point centre is established.
Article 54 is Four Triangle Conics . The construction of the Triplicate Ratio Circle and the 7-Point Circle when the generating point is not the Symmedian point is described. Analysis shows that four (and not just two) Conics emerge. Those that might be termed the Triplicate Ratio Conic and the 7-Point Conic are similar and have the same centre.
Article 55 is Perpendiculars to a Triangle's sides through its Vertices Six points on the sides are generated (other than the feet of the altitudes). If one constructs the three circles through B, C and C, A and A, B that cut the circumcircle orthogonally, the six points are the intersections of these circles with the sides of ABC.
Article 56 is The Cevian point Conic In triangle ABC let P be a Cevian point with D, E, F the feet of the Cevians on BC, CA, AB respectively. Points L, M, N lie on AD, BE, CF respectively and are such that P is the midpoint of AL, BM, CN. It is found that the points A, B, C, L, M, N lie on a conic which we call the Cevian point conic of P. Several other properties are established.
Article 57 is Perpendiculars to the Cevians at the Cevian Point The points where these perpendiculars meet corresponding sides are shown to be collinear.
Article 58 is Additional results for the Miquel Configuration The Miquel point P is the common point of circles AMN, BNL, CLM when L, M, N lie on BC, CA, AB respectively. If perpendiculars to BC at L, CA at M, AB at N are drawn, producing six points on the other sides of ABC, then a variety of results hold. In particular three new circles may be drawn, pairs of which each have a common point with one of the Miquel circles.
Article 59 is 7 Points on any Circle not through a Vertex Three Miquel circles AMN, BNL, CLM with L, M, N on the sides meet at a point P. S is any other circle through P (not through a vertex). AMN meets S at G and X, BNL meets S at G and Y, CLM meets S at G and Z. It is proved that AX, BY, CZ concur at a point lying on S.
Article 60 is Centroid-centred similar ellipses The three ellipses are the outer and inner Steiner ellipses and an intermediate on passing through points a quarter and three quarter away along the sides. Certain sets of collinear points are identified.
Article 61 is Concurrent lines in a triangle with a Circle cutting the Sides A circle cuts the sides of a triangle at the feet of two Cevian points. When this construction is carried out a number of other sets of three lines are concurrent and sets of three points are collinear.
Article 62 is A Triangle with an arbitrary Conic cutting its Sides A conic is chosen that cuts the sides BC, CA, AB of a triangle ABC in points L, U; M, V; N, W (in that order anticlockwise) respectively. The chords LW, MU, NV form a triangle DEF. A large number of concurrencies and collinearities are identified.
Article 63 is A Cascade of Conics A pair of in-perspective triangles are inscribed in a conic. Their non-corresponding sides meet at six points, which turn out to lie on a second conic and form a pair of in-perspective triangles. The process may therefore be repeated indefinitely forming the cascade of conics in the title.
Article 64 is Porism constructed by the Circumcircle and Triangles in Perspective A hexagon is formed by the tangents to the circumcircle at the vertices of the two triangles and a conic passes through the vertices of the hexagon.
Article 65 is When I replaces K and Ge replaces H and Mi replaces O An analogue of the triplicate ratio circle and the 7-point circle is created under these circumstances (Mi is the Mittelpunkt and Ge is Gergonne's point).
Article 66 is On the mean of two Cevian points The feet of their Cevians are used in a construction of a mean that has a geometrical significance.
Article 67 is A Special Tucker Circle The Tucker circle passes through the feet of the perpendiculars to other two sides from the foot of the altitude through their common vertex. The properties of the resulting hexagon are reviewed.
Article 68 is Six Collinear Points in a Special Cyclic Quadrilateral The cyclic quadrilateral has its diagonals at right angles and a review is given of the main properties of such a quadrilateral.
Article 69 is Two Cyclic Quadrilaterals and Two Coaxal Systems It is shown how two cyclic quadrilaterals inscribed in the same circle centre O, each with their diagonals at right angles, generate four circles whose centres lie two by two on lines through O. In each case a coaxal system is generated.
Article 70 is What happens when a Triangle is Rotated about its Orthocentre This is concerned with the properties of two triangles related by a rotation.
Article 71 is More Special Cyclic Quadrilaterals This article is concerned with a cyclic quadrilateral ABCD in which AC is perpendicular to BD and its image A'B'C'D' after a rotation of 90o about E = AC^BD.
Article 72 is What happens when you reflect a Triangle in any Line A triangle ABC and its circumcircle are mapped by a reflection line L with image triangle A'B'C' and its circumcircle. What happens is that the line through A' parallel to BC, the line through B' parallel to CA and the line through C' parallel to AB are concurrent at a point P that always lies on circle A'B'C'.
Article 73 is Constructing Two Coaxal systems by Incidence and Reflection The cyclic quadrilateral ABCD, centre O, with diagonals AC and BD at right angles at E is reflected in the line AC to form the cyclic quadrilateral A' B'C'D', centre O'. From this configuration two systems of coaxal circles are obtained.
Article 74 is The Simson Line Porism Let ABC be a triangle and let A'B'C' be its image under a 180 degree rotation about the circumcentre O. Now let D be any point on the circumcircle S. The locus of the point of intersection of the Wallace-Simson lines of D with respect to the two triangles is an inconic of both triangles, creating a porism.
Article 75 is The Double Simson Line Circle ABC is a triangle and A'B'C' is its image under a rotation by 180 degrees about the circumcentre O. With P a point on the circumcircle the Double Simson lines of P with respect to the triangles are drawn, meeting at a point Q. The locus of Q as P moves round the circumcircle turns out to be the circle centre O.
Article 76 is The Double Simson Line Conic A transversal LMN of a triangle is drawn through the orthocentre, which is the Double Simson line of a point P on the circumcircle. The corresponding Cevian point of the harmonic conjugates of L, M, N is a point Q, whose locus as P moves is a circumconic of ABC.
Article 77 is The Circle centre O and radius OH A number of related results are established about the circle centre the circumcentre O of a triangle ABC and radius OH, where H is the orthocentre of triangle ABC.
Article 78 is More on the Seven-Point CircleMore on the Seven-Point Circle Various properties are established about the Brocard circle and the circles BKC. CKA, AKB through the symmedian point K of a triangle ABC.
Article 79 is Conics in the Ex-circle Configuration From the points of contact of the three ex-circles of a triangle ABC four conics are drawn and their properties are investigated.
Article 80 is Incircle and Excircle Conics A configuration of four conics passing through the points of contact of the incircle and excircles of a triangle ABC is investigated. The intersections and centres of these conics define sets of points whose joins provide sets of concurrent lines.
Article 81 is A Radical Centre that lies on OI The configuration of a triangle ABC with incentre I and the three circles BCI, CAI, ABI is investigated and three new circles are defined whose radical centre is a point J on OI where O is the circumcentre. J is shown to be the isogonal conjugate of Gergonne’s point.
Article 82 is Cevian derived Conics Lines through the feet of Cevians of a point are drawn parallel to the sides of a triangle and meet the sides in six points lying on a conic. Properties of this conic are investigated.
Article 83 is Conics generated by Points on a Curve of degree Five From the feet of Cevians of a point P lines are drawn perpendicular to the sides meeting them in six point. It turns out that these six points lie on a conic if, and only if, P lies on a curve of degree five whose equation is determined. Well known points lying on this curve are O, H, I, the circumcentre, orthocentre and incentre of ABC, and also the excentres.
Article 84 is The G Circles and the Conic they determine From the feet of Cevians of a point P lines are drawn perpendicular to the sides meeting them in six point. It turns out that these six points lie on a conic if, and only if, P lies on a curve of degree five whose equation is determined. Well known points lying on this curve are O, H, I, the circumcentre, orthocentre and incentre of ABC, and also the excentres.
Article 85 is A Converse of the Pascal Line Property It is shown how any transversal of a triangle and any three additional points lying one on each side of the triangle (but not at the vertices) may be used to find six points on the sides of the triangle through which a conic always lies. The proof involves a particular case of the converse of the Pascal line theorem.
Article 86 is The Perpendiculars to Three Segments at a Point determine a Conic It is shown how any transversal of a triangle and any three additional points lying one on each side of the triangle (but not at the vertices) may be used to find six points on the sides of the triangle through which a conic always lies. The proof involves a particular case of the converse of the Pascal line theorem.
Article 87 is Two connected Rectangular Hyperbolae Given a Cyclic Quadrilateral and the Rectangular Hyperbola Σ through the midpoints P, Q, R, S of its sides and its centre O, it is shown that this conic Σ automatically passes through the diagonal points. A slight extension of this famous result is documented.
Article 88 is A Singular Cyclic Quadrilateral When a Cyclic Quadrilateral ABCD is such that the tangents at A and C and the line BD are concurrent, it follows that the tangents at B and D and the line AC are also concurrent. Additional properties of this configuration are obtained.
Article 89 is Perpendiculars in a Cyclic Quadrilateral. Perpendiculars from A and C on to opposite sides of a cyclic quadrilateral ABCD produce four more cyclic quadrilaterals and two sets of parallel lines one set containing five lines and the other set containing six lines.
Article 90 is Perpendiculars from the vertices of a Cyclic Quadrilateral . The perpendiculars from A to AB and DA and the six similar perpendiculars from B, C, D to the sides of a cyclic quadrilateral create two more cyclic quadrilaterals that are coaxal with ABCD. Their sides and their diagonal point lines exhibit some surprising properties.
Article 91 is 6 Conics . When a circumconic is drawn to triangle ABC the tangents at A, B, C form a triangle DEF which is in perspective with ABC. If the lines AD, BE, CF meet at U and meet the circumconic at R, S, T then the tangents at R, S, T form a triangle LMN, It is shown that D, E, F, L, M, N lie on a conic, Then, by drawing tangents and chords four other conics may be constructed with centres all lying on a line through U.
Article 92 is Three special Cyclic Quadrilaterals . In any cyclic quadrilateral ABCD if AB and CD meet at F and AD and BC meet at G then FG is one side of the diagonal point triangle. It is then always the case that the tangents at B and D and the tangents at A and C meet at points U and V respectively lying on FG. The first special cyclic quadrilateral is when AC passes through U. If the tangents at A and B meet at P and Q, R, S are similarly defined then the second special cyclic quadrilateral is when AC is perpendicular to BD and then P, Q, R, S are concyclic. The third special cyclic quadrilateral is when AC is parallel to BD and then again P, Q, R, S are concyclic.
Article 93 is Circles formed by an Isosceles Trapezium . Given an isosceles trapezium ABCD with AD parallel to BC, if tangents to the cyclic quadrilateral ABCD are drawn to produce six points of intersection, then six more circles may be drawn, all passing through the centre O of ABCD. Five of these circles are obvious but the fact that the sixth circle passes through O is an interesting result and in this paper a proof is given using Cartesian co-ordinates.
Article 94 is Generalization of the Steiner Point . The outer Steiner ellipse passes through A, B, C and the images L, M, N of those vertices in a rotation of 180 degrees about the centroid G. The Steiner point is the fourth point of intersection of the outer Steiner ellipse and the circumcircle of ABC. A generalization is obtained by replacing G by another point P internal to the triangle ABC. But more occurs as circles AMN, BMN, CNL intersect at a point U that lies on both the circumcircle and the ellipse ABCLMN and U is the generalized Steiner point.
Article 95 is A Nine Point Rectangular Hyperbola . The configuration consisting of the rectangular hyperbola passing through the incentre, the excentres, the centroid and deLongchamps point in a triangle and three other significant points exhibits some interesting properties which are investigated.
Article 96 is More on the Nine Point Rectangular Hyperbola . The circumcircle of a triangle ABC is the Nine-Point circle of the triangle IJK of its excentres. The line JK passes through A and the midpoint D of JK, which lies therefore on the circumcircle. With E, F similarly defined there are thus two triangles ABC and DEF and it is shown that their incentres and excentres lie on the nine point rectangular hyperbola discussed in CJB/2010/95. Other interesting properties emerge such as the deLongchamps point of DEF is the same point as the incentre of ABC.
Article 97 is How the Excentres create Points on the Circumcircle . The circles through pairs of vertices of a triangle and the excentres opposite the third vertex have centres lying on the circumcircle and pass through the incentre of the triangle. The triangles with these centres as vertices exhibit properties that are described.
Article 98 is Ex-points and their Sets of Intersecting Circles . The ex-symmedian points and the ex-points of an arbitrary point internal to the triangle are shown to lie on a hyperbola. The configuration involving ex-points of a given point and vertices of a given triangle is shown to produce a set of three circles having a point in common (similar to the Fermat point). The relationship between these points is obtained.
Article 99 is A Triangle and its Image under a Half Turn . A triangle ABC and its image DEF under a half turn results in circles BCD, CAE, ABF intersecting at a point P on circle DEF. A point Q on circle ABC is similarly defined. Also it is established that a conic passes through A, B, C, D, E, F, P, Q.
Article 100 is Circles though a point in an Equilateral Triangle . Given three points D, E, F lying on the medians of an equilateral triangle ABC the conditions are determined for the circles BCD, CAE, ABF to have a common point.
Article 101 is Intersections at the vertices of the Second Brocard Triangle . If D, E, F are the ex-symmedian points it is proved that circles BCD, CAE, ABF intersect the Brocard circle at the circumcentre of ABC and at the vertices of the second Brocard triangle. A case of circular perspective is thereby determined.
Article 102 is Two Triangles, their Ex-Symmedians and four Conics . The two triangles are in perspective, vertex the centroid of one of them. The six ex-symmedian points lie on a conic, the tangents at the vertex circumscribe the circumcircle and also the intersections of various lines produce six points lying three by three on two parallel lines. Two other conics of interest are also created.
Article 103 is The Altitudes Create Four Circles, Four Conics and a Polar line . If the altitudes of a triangle ABC meet the circumcircle at D, E, F, then the six interior intersections of the two triangles and the tangents at the above six points produce a configuration in which four circles and four conics feature. An analysis is given in which the polar of the orthocentre plays a major role.
Article 104 is How any Six Points on a Circle create Two Conics . Given three chords of a circle, perpendiculars to those chords from their end points create three pairs of parallel lines. Of their fifteen points of intersection three lie on the line at infinity and the other twelve lie six by six on two conics. Note that the six points must lie on a circle and not a general conic.
Article 105 is External Squares on the Sides of a Triangle . If squares are drawn externally on the sides of a triangle the configuration exhibits two points similar to the Fermat points. Also the six intersections of various lines define a circle.
Article 106 is On Perpendiculars at the corners of a Cyclic Quadrilateral . If perpendiculars to the sides are drawn at the vertices of a cyclic quadrilateral then a configuration is created that consists of two circles coaxal with the first circle, each containing quadrilaterals similar to each other. Their external diagonal points also lie three by three on two collinear lines. Two other circles also emerge.
Article 107 is Conics through three points with Centres on a Fixed Line . Conics through three non-collinear points A, B, C having their centres on a fixed line have three radical axes, BC, CA, AB. If one takes a fixed point P on one of these axes, say BC, then the polars of P with respect to all the conics pass through another fixed point P', which as one would expect, is the harmonic conjugate of P with respect to B and C.
Article 108 is A Property of 3 Circles passing through a fixed Point Christopher Bradley . Let ABC be a triangle and P a fixed point not on the sides. Circle BPC meets AB at W and AC at M. Circle CPA meet BC at U and BA at N. Circle APB meets CA at V and CB at L. The perpendicular bisectors of LU, MV, NW meet at a point Q.
Article 109 is Four Perspectives given a Triangle and its Circumcircle . Given a triangle, its circumcircle and a point not on the circumcircle, it is shown how by varying the position of the point along a line four perspectives may be created.
Article 110 is Three Concurrent Pascal lines and an Auxiliary Conic .
Article 111 is Perpendicular bisectors of Three Radii of a Circle .
Article 112 is Parallels from the feet of Cevians create a Conic . From the feet of Cevians through a point parallels to the other two sides are drawn. The six points created on the sides of the triangle lie on a conic.
Article 113 is Circles passing through the Miquel Point of the feet of Cevians . If LMN are the feet of a set of Cevians, then circles AMN, BNL, CLM meet at a Miquel point Q. If now circles BQC, CQA, AQB are drawn and circles AMN and BQC meet at R with S, T similarly defined, then circle RST passes through the Miquel point Q. The centres of the seven circles also exhibit some remarkable properties showing Q to be the Miquel point of a second triangle, a cascade process that can be carried on indefinitely.
Article 114 is Properties of a Particular Tucker Circle . Three circles determine a particular Tucker circle and their centres exhibit some interesting properties.
Article 115 is Six Circles and their Centres . In triangle ABC centroid G, circles BGC, CGA, AGB are drawn meeting the sides in six points that lie on a conic. Three more circles are drawn and the six circle centres exhibit some interesting properties.
Article 116 is The Geometry of the Brocard Axis and Associated Conics . From a triangle and its symmedian point K and its circumcentre O, various triangles, the Brocard inellipse and two further conics are constructed having their centres on the Brocard axis KO. Two porisms are defined and a special line is identified. Most, if not all, of the content of this article is known and indeed should be known by all geometers.
Article 117 is When the Cevians of triangle ABC meet the Circumcircle at D, E, F . In a triangle ABC when the Cevians through the centroid G meet the circumcircle Σ at points D, E, F and the tangents are drawn at the six points, two triangles are created. The intersections of the sides of the four triangles in the configuration have some interesting properties.
Article 118 is Two Triangles in Perspective and inscribed in a Conic . Given two triangles ABC, DEF in perspective at P and inscribed in a conic Σ, a configuration is produced consisting of the 6 Pascal lines and 2 Steiner points arising. One of the Pascal lines has further properties, as it is not only a Pascal line, but also a Desargues’ axis of perspective and also the polar of P with respect to Σ. Further properties involving the nine lines AD, AE, AF, BD, BE, BF, CD, CE, CF and the tangents at the six vertices are deduced leading to a conic, further properties of the key Pascal line and numerous sets of collinear points and concurrent lines.
Article 119 is The Nine-point Conic and a Pair of Parallel Lines . An affine transformation of the nine-point circle leads to the nine-point conic. The orthocentre transforms into a general point P which is collinear with the centre of the nine-point conic and the centroid. We prove that the polar of P with respect to the nine-point conic is parallel to the Desargues’ axis of perspective of the triangle and the image of the orthic triangle.
Article 120 is An Important Line through the Centre of a Cevian Inellipse . A Cevian inellipse is one that touches the sides of a triangle internally at points that are the feet of Cevian lines. It is shown in this short article that the line through the Cevian point and the centre of a Cevian inellipse contains other significant points.
Article 121 is Some Lines through the Centroid G . Using areal co-ordinates we catalogue some lines through the centroid G of a triangle ABC. Co-ordinates of points and equations of lines are given.
Article 122 is Some lines through the Incentre I . Using areal co-ordinates we catalogue some lines through the incentre I of a triangle ABC. Co-ordinates of points and equations of lines are given.
Article 123 is Properties of the Intangential Triangle . Triangle ABC and its incircle and ex-circles are given. Common tangents to the incircle and an ex-circle are four in number; three being the sides of ABC and the fourth is called an intangent. There are three intangents, one for each ex-circle and these form a triangle A'B'C' called the intangential triangle. Properties of the intangential triangle are studied using areal co-ordinates with ABC as triangle of reference. By construction its incircle is the same as that of triangle ABC. The triangle of its excentres has sides parallel to those of ABC.
Article 124 is Two Cevian Points Collinear with a Vertex and Thirteen Conics . Given triangle ABC, a point P on BC and lines LMP and UVP with L, U on AB and M, V on CA, then two Cevian points may be constructed collinear with vertex A. Two further Cevian points occur and the resulting configuration of points and lines results in the appearance of thirteen conics and two harmonic ranges.
Article 125 is Properties of the Extangential Triangle . The external common tangents to the three ex-circles form the extangential triangle A'B'C'. Triangles ABC and A'B'C' are in perspective with perspector the orthocentre of the intouch triangle. The axis of perspective plays a prominent part in the configuration. Triangle A'B'C' and the orthic triangle are homothetic through the Clawson point.
Article 126 is Some Properties of the Ex-Circle configuration . The ex-circle configuration is studied and it is shown in particular how the midpoints of the sides of the ex-central triangle lead to several circles and perspectives, as well as a new (to me) triangle with vertices on the sides of the original triangle and two new (to me) points on the circumcircle.
Article 127 is When a Circle passes through a Vertex and cuts the Opposite Side . A surprising configuration emerges when a circle passes through a vertex and cuts the opposite side. Two pairs of touching circles and two circles each passing through seven key points emerge and additional points on the original circle are also created. It seems unlikely that a synthetic solution will exist, as some of the co-ordinate working is technically very difficult.
Article 128 is The Symmetry of a Scalene Triangle . Points dividing each side of a scalene triangle in a fixed ratio leads to a configuration with two sets of three equal circles and a conic. Many results are self-evident but the equations of the circles and conic are recorded.
Article 129 is The Neuberg Cubic . Analysis is made of the Neuberg triangle cubic. This is the locus of points P with the property that the reflection of P in the sides of a triangle ABC leads to a triangle which is in perspective with ABC.
Article 130 is An Extension to the Theory of Hagge Circles . An extension of the theory of Hagge circles is presented.
Article 131 is Construction of Circles always having centre the Nine-Point Centre . From any general point P reflections in BC. CA, AB give D, E, F. The midpoints of AD, BE, CF are U, V, W respectively. U, V, W are shown to lie on a circle with centre the Nine-Point centre.
Article 132 is The Cevian Conic . If P is a Cevian point in a triangle ABC and D, E, F are the feet of the Cevians, then three circles PEF, PFD, PDE meet the sides again in six other points. It is shown that these six points lie on a conic, which we call the Cevian Conic.
Article 133 is Conics in a semi-regular hexagon. Hexagon AFBDCE with ABC an equilateral triangle and AD, BE, CF concurrent at the centroid P of ABC is inscribed in a conic, and as such is defined as a semi-regular hexagon. It is proved that the six circumcentres of triangles AFP, FBP, BDP, DCP, CPE, EPA are co-conic as are their six orthocentres. It is also proved that the intersections of contiguous Euler lines are co-conic. Cabri indicates that the six in-centres and six nine-point centres are also co-conic.
Article 134 is When the Cevian Conic is a Circle . When the Cevian point is Gergonne's point, the Cevian Conic is a Circle with centre at the incentre.
Article 135 is Constructing Triangles with Coincident Centroids . Given two triangles with coincident centroids it is shown how to construct an unlimited number of triangles with the same centroid.
Article 136 is Properties of the symmedian point in a cyclic quadrilateral. If ABCD is a cyclic quadrilateral then the symmedian points of triangles ABC and DBC are shown to have some remarkable properties. Many pairs of lines may be constructed whose intersections lie on the sides of the diagonal point triangle.
Article 137 is Affine tranform of the properties of the Eulerian triangle. Given a triangle ABC and a point K a configuration is constructed containing a triplicate ratio conic, a circumconic, a nine-point conic and a 7 point conic. When K is actually the symmedian point K(a2 b2, c2), then these conics are the familiar circles associated with a triangle in the Euclidean plane. In this article we take K, a point not on the sides of ABC, to be an arbitrary point K(f, g, h) and what emerges is an affine map of the Euclidean plane.
Article 138 is Explaining some collinearities among triangle centres. Collinearities are preserved by an Affine Transformation, so that for every collinearity of triangle centres, there is another when an affine transformation has taken place and vice-versa. We illustrate this by considering what happens to three collinearities in the Euclidean plane when an affine transformation takes the symmedian point into the incentre.
Article 139 is The harmonic cevian conic. If D, E, F are the feet of Cevians with D on BC etc. and if L is the harmonic conjugate of C with respect to B and D and U is the harmonic conjugate of B with respect to C and D and if M, N, V, W are similarly defined then a conic passes through the six points L, U, M, V, N, W. There is one internal Cevian point for which the conic is a circle.
Article 140 is The centroid of centroids. Let P be a point not on the sides of a triangle and suppose the centroids of triangle PBC, PCA, PAB are L, M, N respectively. Now let J be the centroid of triangle LMN. Then J lies on line PG, where G is the centroid of triangle ABC. The point of concurrence Q of AL, BM, CN also lies on PG.
Article 141 is Three centroids created by a cyclic quadrilateral . The centroid of the quadrilateral considered to be an area of constant density is G, the centroid of the quadrilateral considered as having unit masses at its vertices is N, the centroid of the quadrilateral considered as having unit masses at its vertices and mass of two units at E (the intersection of its diagonals) is F. It is shown that E, F, N, G are collinear.
Article 142 is Incircle conjugation . A construction is presented in which points (f, g, h) in barycentric co-ordinates are transformed into points (a/f, b/g, c/h). This is clearly a conjugation and we call it incircle conjugation.
Article 143 is The nine-point circle of the diagonal point triangle . An analysis is conducted of the ten point rectangular hyperbola through the mid-points of a cyclic quadrilateral and of the nine-point circle of the diagonal point triangle.
Article 144 is Four nine-point circles . In any quadrilateral ABCD the four nine-point circles of triangles BCD, ACD, ABD, ABC have a point in common that lies on the hyperbola through the mid-points and on the circumcircle of the diagonal point triangle.
Article 145 is Concyclic circumcentres in the Steiner configuration . In triangle ABC with centroid G, the points D, E, F lie on AG, BG, CG respectively and are such that AG = GD, BG = GE, CG = GF. It is proved that the circumcentres of triangles AFG, FBG, BGD, DGC, CGE, EGA are concyclic.
Article 146 is More conics in a semi-regular hexagon . ABC is an equilateral triangle and AFBDCE is a hexagon inscribed in a conic. Triangles ABC and DEF are in perspective with vertex P, the centroid of ABC. G1-G6 are the centroids of triangles EAF, FBD, DCE, BDC, CEA, AFB. It is shown that these points lie on a conic. Also triangles G1G2G3 and G4G5G6 are congruent and in perspective. The six interior points of intersection of triangles ABC and DEF also lie on a conic.
Article 147 is The midpoint rectangular hyperbola again . If A, B, C, D are four points on a circle and E is any other point, then the locus of the centre of conic ABCDE as E varies is the rectangular hyperbola through the midpoints of the quadrangle ABCD.
Article 148 is A circle through two vertces, three circumcentres and a Miquel point . Starting with a triangle ABC and a point D on BC it is shown how the circumcentres of triangles ABC, ADB, ADC lie on a circle through A. If this circle intersects circle ABC at A and P, then it is shown that triangle PBC exhibits a Miquel point Q with points D, M, N on its sides.
Article 149 is Circles with a common point in a cyclic quadrilateral If ABCD is a cyclic quadrilateral it is possible to find points P, Q, R, S, T, U on sides AB, BC, CD, DA, AC, BD respectively so that circles BPQU, APST, CQRT, DRSU have a common point M. These circles have centres that are concyclic, as are points M, U, I, T where I = AC^BD.
Article 150 is When a point is a circumcentre, an incentre and an orthocentre . Given a triangle ABC, circumcentre O, and the centres D, E, F of circles BOC, COA, AOB respectively, then O is the incentre of triangle DEF, as well as being the orthocentre of the triangle of excentres.
Article 151 is The Orthocentre Conics . When circles BHC, CHA, AHB in a triangle ABC with orthocentre H are drawn their centres D, E, F have interesting properties.
Article 152 is Generating Circles from the Symmedian point . Given a triangle ABC and its symmedian point K circles BKC, CKA, AKB are drawn. Their other points of intersection with the sides of ABC are shown to lie on a Tucker circle. These points also lie in pairs on the sides of another triangle and the other intersections of these sides with the circles BKC, CKA, AKB are also concyclic. Centres of these circles are shown to lie on the Brocard axis.
Article 153 is More properties of the Incentre . In a triangle ABC, with incentre I, the midpoints of AI, BI, CI are located. Perpendicular bisectors of AD, BE, CF are drawn. These meet the interiors of the sides of ABC at six points that are shown to lie on an ellipse. As the figure develops to include the excentres several other circles and conics are located.
Article 154 is On the Incircle and Excircles of a Cevian Triangle . Given triangle ABC, if D, E, F are the feet of the Cevians AD, BE, CF and L, M, N are the points of contact of the incircle of triangle DEF with the sides of triangle DEF then AL, BM, CN are con ncurrent, as are AL', BM', CN', where L', M', N' are the points of contact of the ex-circles with EF, FD, DE respectively.
Article 155 is The Circumconic of a pair of Cevian Triangles . In triangle ABC two Cevian triangles are drawn and their points of intersection and the vertices of ABC lead to six sets of collinear points. Several conics are identified.
Article 156 is Circumcentre Conics . Given triangle ABC, let triangle UVW be a reduction of ABC by an enlargement factor t (0 < t < 1) about the circumcentre O. The sides of triangle UVW cut the sides of ABC in six points that lie on a conic. It is shown that the centres of all such conics lie on the Brocard axis.
Article 157 is The General Inellipse . The general inellipse in a triangle touches its sides at the feet of Cevians. An analysis is given that relates the Cevian point to the centre of the inellipse and to two related circumconics.
Article 158 is Six Point Circles and their Associated Conics . If P is a point internal to a triangle ABC and AP, BP, CP meet the circumcircle again at A', B', C' respectively and U, V, W are the midpoints of AP, BP, CP and D, E, F are the midpoints of PA', PB', PC' respectively then U, V, W, D, E, F are obviously concyclic. But the configuration has other properties which are pointed out.
Article 159 is A Twelve Point Configuration and Carnot’s Theorem . A twelve point configuration occurs when six of the points lie on a conic and the remaining six points are the vertices of two triangles in perspective. In this article we consider the configuration in which a conic cuts the three sides of a triangle in real points. A number of theorems create a variety of conditions on the points of intersection equivalent to Carnot’s theorem.
Article 160 is Tangents to a Conic from the vertices of a Triangle . Given a conic which cuts the sides of a triangle in six real points, tangents A1, A2, B3, B4, C5, C6 are drawn to the conic. The tangents meet each other at 12 points (15 if you count A, B, C). The se form a 12 point configuration (see Article 159) so that six of the twelve points lie on a conic.
Article 161 is When 12 points display eight 6-point Conics and 6 Concurrent Lines . Six lines being tangents to a circle at vertices of triangles in perspective create twelve points of intersection. Six conics with six points each can be drawn through the twelve points, each point lying on three of the conics. Also six lines can be drawn through the twelve points, lines which are concurrent at the vertex of perspective.
Article 162 is When 24 Points form three 8-point Conics and 12 Concurrent Lines . A construction is described involving two chords of a conic that results in 24 points that form three 8-point Conics and 12 concurrent Lines.
Article 163 is A Theorem on the Complete Quadrilateral . Given a complete quadrilateral ABCDEF and two points P and Q then it is well-known that conics ABCPQ, AFEPQ, BFDPQ, and CDEPQ have a common point R. It is proved that when P, Q are the midpoints of the diagonals BE, CF respectively, then R lies on the third diagonal AD.
Article 164 is The Transversal of a Quadrilateral . A transversal LMNP intersects a convex quadrilateral ABCD with L on AB etc. The harmonic conjugates of L, M, N, P are L', M', N', P'. It is proved that L'P' and M'N' intersect on the diagonal BD and that L'M' intersects N'P' on the diagonal AC.
Article 165 is A description of properties of Pascal's hexagon . A description is given of properties of Pascal’s hexagon, illustrating how 6 Pascal lines fall into two sets of 3 concurrent lines, defining 2 Steiner points.
Article 166 is Analytic treatment of a Romanian problem. . A triangle ABC is given and a conic that cuts each of its sides in two real points. The tangents at these points are drawn providing a hexagon of tangents circumscribing the conic, whose vertices are labelled DF'ED'FE as in Fig.1. It is shown that AD, BE, CF are concurrent as are AD', BE', CF' and DD', EE', FF'. The positions of the resulting perspectives P, P' and Q are located. In the Romanian question the conic was the nine-point circle and then P lies on the hyperbola ABCKOi, where K is the symmedian point and Oi is the isotomic conjugate of the circumcentre O. The isotomic conjugate of P then lies on the diameter of the circumcircle joining the Steiner point and the Tarry point. It is also shown that in the general case AD', EF', FE' are concurrent as are BE', DF', FD' and CF', DE', ED'. Areal co-ordinates are used throughout with ABC the triangle of reference.
Article 167 is Basic Properties of a Quadrangle possessing an Incircle . Properties of a quadrangle possessing an incircle.
Article 168 is Properties of a pair of Diametrically Opposite Triangles . Starting from a triangle ABC and its circumcircle centre O, triangle DEF is such that AD, BE, CF pass through O. Tangents at the six points and the nine lines joining their vertices create a configuration that is explored in this article.
Article 169 is Two In-Perspective Triangles inscribed in a Conic . Triangles ABC and DEF are inscribed in a conic and are in perspective through a point O. Polar lines LMN and PQR are drawn and six points of concurrency are shown to define a number of important collinearities. Two conics also emerge.
Article 170 is The Brocard Conics . Triangles ABC and DEF are inscribed in a circle and AD, BE, CF are concurrent at the symmedian point K. Sides of the two triangles are extended and the tangents at the vertices are drawn. The polar lines of the two triangles coincide and points of intersection of the sides and tangents are shown to produce in addition four six-point conics all with their centres on the Brocard axis OK, where O is the circumcentre of ABC.
Article 171 is A Conic through the feet of two Cevians may lead to a second Conic. Points P and Q are two Cevian points so that the feet U, V, W, U, V', W' carry a conic then the chords UV, WV', W'U' intersect as shown if and only if ABCDEF is a conic. Other properties (not proved in this article) hold as may be inferred from Fig. 1.
Article 172 is The Inner Circle of a Cyclic Quadrilateral. Given a cyclic quadrilateral ABCD with midpoints of sides P, Q, R, S and midpoints of diagonals U, V, then it is well-known that a rectangular hyperbola passes through P, Q, R, S, U, V as well as O, the centre and H, J the other two diagonal points. We find the equation of this hyperbola and that of the inner circle OUVT, centre G. The midpoint W of UV is also the midpoint of PR and of QS.
Article 173 is The Auxiliary Circles of a Cyclic Quadrilateral. First the cyclic quadrilateral ABCD is drawn. The midpoints of AB, BC, CD, DA, AC, BD are labelled P, Q, R, S, U, V, the centre of ABCD is the point O and the diagonal points are T = AC∧CD, H = AB∧DC and K = AD∧BC. (These are all finite points.) The 10 point rectangular hyperbola P, Q, R, S, O, U, V, T, K, H is shown. We establish the existence of the three midpoint circles OUVT, OPRH, OQSK and also the Semi-Diagonal point Circle LMNXYZ, where L, M, N are the midpoints of OT, OH, OK and X = OK∧HT, Y = OH∧KT and Z = OT∧H.
Article 174 is A Cevian Circle leads to Perspective Triangles. A circle is drawn through U, V, W the feet of a Cevian point T. Three other points are created on the sides of triangle ABC enabling a second triangle DEF to be drawn. It is shown that triangles ABC and DEF are in perspective.
Article 175 is When two Triangles intersect in Conics and in Circles. If P is internal to a triangle and points D, E, F are defined on AP, BP, CP respectively so that AD/AP = BE/BP = CF/CP then the intersections of triangles ABC and DEF lie on a conic. If P is the symmedian point K then the conic is a circle.
Article 176 is Orthologic Triangles. Triangles ABC and DEF are orthologic if the perpendiculars from the vertex of one of them to the corresponding sides of the other are concurrent. The property is symmetric but not transitive.
Article 177 is Special Pascal Lines. By considering the hexagon as two triangles in perspective and labelling their vertices as well as those of the defining conic, it is shown that two of the Pascal lines pass through points that are points of concurrence of three lines.
Article 178 is More Circles centred on the Brocard Axis. Circles through B and C and the Brocard points lead to two circles centred on the Brocard axis and several concurrences.
Article 179 is The Second Brocard Triangle. Given the Brocard points Ω, Ω' in a triangle the point of concurrence of the intersection of the line AK with the 7-point circle and circles AΩC, BOC, AΩ'B (D in the above diagram) and two similar points E, F found by cyclic change of A, B, C forms what is known as the Second Brocard Triangle (David Monk, private communication). An analysis is the content of this document.
Article 180 is Circles through the Brocard points and the Circumcentre. With Brocard points Ω and Ω', circles are drawn through AΩΩ', BOW', COW and they meet at a point D on the circumcircle. Points E, F are similarly defined. It is proved that the lines AD, BE, CD are all parallel to ΩΩ'. Further, if circle AΩΩ' meets AB at X and AC at Y then XY is also parallel to ΩΩ', as are two further lines defined in a similar fashion to XY by cyclic change.
Article 181 is Circles through the Brocard points and the Symmedian Point. Circles BΩK, CΩK both pass through the same point D on BC. E and F are defined by cyclic change and lie on CA and AB. DEF turn out to lie on a line parallel to the tangent at K to the 7-point circle (and hence perpendicular to the Brocard axis).
Article 182 is Problems requiring Proof. Here are four problems. In fact the solution to problem 4 is known, but the other problems may well be open.
Article 183 is Conics and Triangles in Perspective. It is shown how a pair of triangles in perspective lead to a conic and conversely.
Article 184 is Cevian Perspectivity. Triangle ABC and PQR with Cevian points D and S respectively are said to be in Cevian perspective if the points 11 = AD∧PS, 22 = BD∧QS, 33 = CD∧RS are collinear. If points are labelled as follows: 12 = AD∧QS,13 = AD∧RS, 21 = BD∧PS, 23 = BD∧RS, 31= CD∧PS, 32 = CD∧QS, then the following results hold: (i) triangles 13 23 21 and 31 32 12 are in (ordinary) perspective through a vertex X, (ii) the points 13 23 21 31 32 12 are co-conic and (iii) X lies on the line SD.
Article 185 is The Four Conic Theorem. Given two sets of four lines which intersect one on one in four points in a straight line, then the twelve remaining points carry four conics in which through point two conics pass.
Article 186 is 6 Points and 4 perspectives If 6 points provide 1 pair of triangles in perspective then there are 3 more pairs of perspective triangles, with the same vertex, in the same diagram. Their Desargues' axes create the sides of a triangle and a related transversal. (It is assumed the original pair of triangles are not in triple perspective.)
Article 187 is A Perspective in a pair of Cyclic Quadrilaterals. When two cyclic quadrilaterals in the same circle are in perspective then an axis of perspective joining the intersections of all six pairs of corresponding sides exists.
Article 188 is Are three Triangles ever in Mutual Perspective? The question in the title has the answer `Yes', for example three triangles in the Brocard porism are in fact in triple reverse perspective with each other. However, we study in this very short article the condition that three triangles must obey if they are to be in mutual perspective. (Perspectives, of course, are symmetric, but not generally associative.)
Article 189 is The Midpoint Conic. It is familiar that the feet of a pair of Cevians are co-conic. We prove here that their midpoints are also co-conic.
Article 190 is The Brocard Circles' Twin Conic The construction of the Triplicate Ratio Circle and the Brocard Circle is replicated but starting with the circumcentre O rather than the symmedian point K. It is found that the Brocard Circle is replaced by a conic (the twin conic) passing through O and K and having centre the midpoint of OK. This result leads to the concept of a conjugation whose properties we briefly outline.
Article 191 is Three Circles and their Centres Given a cyclic quadrilateral and a point P not on a side, lines may be drawn through P parallel to its sides each line meeting the adjacent sides in a point. The eight points created in this way determine two circles and the three circle centres and P form a parallelogram.
Article 192 is Generalization of the Triplicate Ratio Circle Given a triangle ABC a point P not on its sides is selected and lines parallel to AB and AC are drawn through P to meet the sides in four distinct points. The position of P is determined when these four points are concyclic.
Article 193 is A Two Circle Problem Involving the GK axis Through a point X lines are drawn parallel to the sides of ABC. From the resulting six points two circles are drawn and the condition on X for their common chord to pass through X is determined, the result being that X must lie on the GK axis.
Article 194 is The KG Nine-Point Conic The properties of a Nine-Point Conic are exhibited using Cevian point K (as well as the centroid G). The symmedian point Km of the medial triangle (X141) is shown to lie on the KG axis, the midpoint of KKm being the centre X of the nine-point conic.
Article 195 is The Orthocentre Circle of a Cyclic Quadrilateral With ABCD a cyclic quadrilateral perpendiculars are drawn from A to BC and to CD meeting those sides at L and N respectively and perpendiculars are drawn from B to DA and to CD meeting those sides at Z and X respectively. Further the perpendiculars from C to AB and DA are drawn meeting those sides at M and P respectively and finally perpendiculars are drawn from D to AB and BC meeting those sides at Y and W respectively. Point A' = BZ^DY, point B' = CM^AL, point C' = BX^DW and point D' = AN^CP. It is found that A', B', C', D' are concyclic and The circle we term the orthocentre circle as it mirrors the construction of the orthocentre in a triangle. Nine other circles arise out of the figure and their properties are investigated.
Article 196 is The Two Central Lines in a Cyclic Quadrilateral There are five main central points in a cyclic quadrilateral: the circumcentre O, the centroid F, the centre of mass G, the intersection of the diagonals E and the intersection of the lines joining the midpoints of opposite sides T. It is proved in this article that GTE is a straight line and that OTF is a straight line. Furthermore ET = 3TG and OT = 3TF and so GF is parallel to OE. (The anticentre also lies on OTF.)
Article 197 is When Two Pairs of Diagonals are Concurrent If ABCD is a cyclic quadrilateral and the tangents at A, B, C, D form a quadrilateral PQRS, the diagonals AC, BD and PR, QS all pass through a given point. Here we identify this point.
Article 198 is The Two Cevians' Perspective In a triangle Cevians are drawn through P and Q. The points L, M, N, R, S, T are defined as follows: L = BQ^CP, M = BQ^AP, N = AQ^CP, R = CQ^BP, S = AQ^BP, T = CQ^AP. It is proved that triangles LMN and RST are in perspective.
Article 199 is Three Circles with collinear Centres With ABCD a cyclic quadrilateral centre O and midpoints of sides PQRS, lines PO, QO, RO, SO meet the opposite sides at T, U, V, W respectively. Circles SUQW and TVPR have centres F and E respectively. It is proved that FOE is a straight line. Some other obvious results are mentioned.
Article 200 is Problem on Quadrilateral with an Incircle Given a quadrilateral with an incircle there are vertices A, B, C, D and points of contact P, Q, R, S. Lines AQ, AR, CP, CS are drawn intersecting in K and M. Lines BS, BR, DP, DQ are drawn intersecting in N and L. It is proved that K, M lie on BD and N, L lie on AC.
Article 201 is All about a Cyclic Quadrilateral and its cousin Given a Cyclic Quadrilateral ABCD with circumcentre O and diagonals AC, BD, EF then the midpoints of the diagonals P, Q, R respectively are well known to be collinear. The points U, V, W are the intersections of the diagonals. It is first shown that circle UPQ passes through O and that if the line RU meets this circle at S, then the following circles all pass through S. These circles are VRP, WRQ, UVW, EFO, ACR, BDR. If the circles WVA, WVB, WVC, WVD meet the circle ABCD at points A', B', C', D' then RAA', RBB', RCC', RDD' are all straight lines. It is not unreasonable to say that S is the most important point in the figure.
Article 202 is The Midpoint Theorem Let ABCD be a quadrangle, with L, M, N the midpoints of AB, BC, CA and U, V, W the midpoints of AD, BD, CD, then the conics AULMN, BVLMN, CWLMN have a fourth point in common. (D must not lie on the medians of ABC.)
Article 203 is Another Circle with Centre on the Brocard axis In triangle ABC with symmedian point K, circumcentre O, circles BKC, CKA, AKB meet the sides at six further points, two on each side. It is shown these points lie on a circle with centre at a point O' on the Brocard axis such that OK = 2KO'.
Article 204 is Between the Triplicate Ratio Circle and the Circumcircle Article 205 is The Brocard Lines KΩ and KΩ' Circles A&OmegaΩ', B&OmegaΩ', C&OmegaΩ' meet the sides of ABC also in six further points which lie on two lines K&Omega, KΩ' (which we call the Brocard lines). These six points also lie on four circles, each of the six points lying on two of the circles.
Article 206 is Dividing OH into Five or Seven Equal Parts By dividing each side of a triangle into quarters and then drawing the resulting nine Cevian lines, six points are created internally through which two circles are drawn. Their centres lie on the Euler line and divide OH in the ratios 1:4 and 4:3.
Article 207 is The Median Conic Given a triangle ABC and its centroid G, the midpoints of AG, BG, CG are denoted by U, V, W. Conics BCUVW, CAUVW, ABUVW are drawn and meet the sides again at point D, D', E, E', F, F'. It is proved that D, D', E, E', F, F' lie on a conic.
Article 208 is A Pair of Isogonal Conjugates produce a Conic through 6 Points If P and Q are an isogonal conjugate pair then circles APQ, BPQ, CPQ intersect the sides of triangle ABC in six points and it is found that a conic passes through them. The construction only produces a conic when P and Q are isogonal conjugates. When P and Q are the Brocard points then the conic degenerates into a pair of lines.
Article 209 is Three Concentric Circles Starting with a cyclic quadrilateral ABCD with centre O, the midpoints A', B', C', D' of OA, OB, OC, OD respectively form a second cyclic quadrilateral with half the size. It is shown that points B, B', C', C lie on a circle and so do C, C', D', D etc. The centre of BB'C'C is labelled bc, and that of CC'D'D is labelled cd etc. Five pairs of these circles are immediately seen to be coaxal and analysis is given for one such pair.
Article 210 is 12 Points, 8 six-point Conics, 4 Conics through each Point Given a triangle ABC from the midpoints of each side perpendiculars are drawn to the other two sides. These lines are labelled Lines 1 to 6 as in the Figure. Their intersections create 12 finite points, so that, for example 35 is the intersection of Lines 3 and 5. It is found that 8 six-point conics can be drawn through these 12 points, with 4 of the conics through each point. The nine-point circle is, of course, one of the conics. In a later article some properties of these conics are established.
Article 211 is A cyclic quadrilateral and its midpoint circles A cyclic quadrilateral its midpoints and The Six Midpoint Circle
Article 212 is Quarter size Circles at Triangle Vertices A cyclic quadrilateral ABCD with AC perpendicular to BD is such that the tangents at A, B, C and D form a cyclic quadrilateral. This configuration is described analytically. If AC^BD = X and the centres of the two circles are O and Y, then it is proved that YOX is a line.
Article 214 is The Super-Cevian Triangles, their Conic and Two Perspectives In the above Figure, ABC is a triangle, P is a Cevian point and D, E, F the feet of the Cevians on BC, CA, AB respectively. The triangle DEF is the Cevian triangle. A triangle XYZ is constructed by drawing through A a line parallel to EF and through B and C lines parallel to FD and DE respectively. Also a triangle UVW is constructed by drawing lines parallel to EF, FD, DE through D, E, F respectively. These two triangles we term as Super-Cevian triangles. We then prove that ABCUVW is a conic. Q is the perspector of triangles UVW and XYZ and R is the perspector of triangles ABC and XYZ. These results are established in this article, using areal co-ordinates with ABC as triangle of reference.
Article 215 is Multiplication of Points using Barycentric Co-ordinates If you have points with barycentric co-ordinates (f, g, h), (u, v, w) then the rule of multiplication is that the result has co-ordinates (fu, gv, hw). It is shown in this article how to perform this multiplication using a geometric construction . The method is as follows: First draw the circumconic fyz + gzx + hxy = 0. Next take the isotomic conjugation of (u, v, w) to get the point (1/u, 1/v, 1/w). To conjugate this point finally use the circumconic to perform the second conjugation taking (1/u, 1/v, 1/w) to the product point (f/(1/u), g/(1/v), h/(1/v)) = (fu, gv, hw). This is evidently a commutative product and so the product may also be obtained using the conic ux + vy + wz = 0 and operating the two conjugations on the point (f, g, h).
Article 216 is A Point, two Triangles and Two Conics A triangle ABC is given along with its circumcircle Γ and any internal point P. Lines AP, BP, CP are extended to meet Γ at U, V, W respectively. Triangle UVW is drawn and their sides intersect internally in six points. BC meets AU at a, with b and c similarly defined. AU meets side VW at u with v and w similarly defined. It is shown in this article that a conic passes through the first six points and that a conic also passes through a, b, c, u, v, w.
Article 217 is An interesting Perspective in the Anticomplementary Triangle A triangle ABC and its anticomplementary triangle A' B'C' are drawn and a point P is selected through which Cevians A'P, B'P, C'P are drawn meeting the sides of ABC in points L', M', N' and the sides of A', B', C' in points L, M, N. Circles LMN and L'M'N' are drawn meeting the sides of A'B'C' and ABC respectively in points U', V', W' and U, V, W. Several perspectives are created, but the most interesting is that of triangles A'B'C' and UVW with perspector Q.
Article 218 is Properties of the Triangle of Excentres In the triangle of excentres the orthocentre is the incentre I of ABC, the Symmedian point K+ is the Mittenpunkt of ABCand the nine-point centre is the circumcentre O of ABC. The circumcentre O+ lies on the line OI and is such that O+O = OI. These properties are proved as also the fact that KI passes through K+.
Article 219 is The Twelve point Circle In the triangle of excentres the orthocentre is the incentre I of ABC, the Symmedian point K+ is the Mittenpunkt of ABCand the nine-point centre is the circumcentre O of ABC. The circumcentre O+ lies on the line OI and is such that O+ O = OI. These properties are proved as also the fact that KI passes through K+. Wow!
Article 220 is A Conical Hexagon with Main Diagonals Concurrent A hexagon ABCDEF is inscribed in a conic with diagonals AD, BE, CF concurrent at a point P. It is shown that this property is replicated and that both conics have the same polar with respect to P. This polar is one of the Pascal lines.
Article 221 is Four Special Conical Hexagons all with same Polar Given a hexagon ABCDEF inscribed in a conic with AD, BE, CF concurrent at P, the tangents tA, tB, tC, tD, tE, tF are drawn. Points jk = tj^tk are determined and it is shown that 12, 23, 34, 45, 56, 61 lie on a conic. Lines 61 12, 34 45 meet at point 1, lines 12 23, 45 56 meet at 2 and lines 23 34, 56 61 meet at 3. Points 1, 2, 3 are collinear and it is shown that 123 is the polar of P with respect to both conics.
Article 222 is The Missing Point on the Euler Line Given a triangle ABC, circumcentre O, orthocentre H, let the midpoints of sides be L, M, N and suppose AO, BO, CO meet the sides BC, CA, AB respectively at U, V, W then the centre of the ellipse LMNUVW is a point S on the Euler line such that OH = 4OS. There appears to be no reference to this point in the literature, and so we call it The Missing Point.
Article 223 is When a non-regular Cyclic Pentagon leads to Another It can be arranged that a non-regular cyclic pentagon ABCDE, by special adjustment of D and E, produces a second cyclic pentagon A'B 'C'D'E', where A' = BD^CE, B' = AC^DE, C'= AB^DE, D'= AB^CE and E' = AC^BD.
Article 224 is The Sixty Pascal Poles When six points lie on a conic then sixty Pascal lines may be constructed. In this paper it is shown how the poles of sixty these lines may be constructed without drawing tangents. Their properties are, of course, the duals of those of the Pascal lines.
Article 225 is A Typical Pascal line Drawing ABCDEF is a cyclic hexagon (though the results remain true if the circle is replaced by a conic). A'B'C' is a Pascal line, where A' = BC^AF, B' = CD^AF and C' = DE^AB. We denote this by (AEC, DBF), where care must be taken with the order of letters. Points X, Y, Z are given by X = AC^DF, Y = AE^DB, Z = EC^BF. Points X', Y', Z' are given by X' = AD^CF, Y' = AD^EB, Z' = CF^EB. In Article 225 it is shown that XX', YY', ZZ' are concurrent at a point P which is the pole of A'B'C' with respect to the circle.
Article 226 is Special Conical Hexagons Hexagon ABCDEF inscribed in a Conic is said to be special if AD, BE, CF are concurrent at a point P. When this happens hexagons formed by taking the intersections of neighbouring tangents are proved also to be special and conical with the same point P.
Article 227 is The Remarkable Eight-Point Conic Two conics intersect at A, B, C, D and AC meets BD at P. A line through P is drawn meeting one of the conics at F, G and the other at E, H. Tangents are drawn at F, G, E, H. Tangents at F and G meet at Q and tangents at E and H meet at R. Tangents at H and G meet at V, tangents at H and F meet at S, tangents at E and F meet at T and tangents at E and G meet at U. A conic now passes through the 8 points A, B, C, D, S, T, U, V and the line QR is the polar of P with respect to all three conics. The general case is technically too difficult to establish, so here we provide a numerical case.
Article 228 is How 2 Conics through 4 Points generate 6 more such Conics Two Conics through 4 point A, B, C, D are drawn (in the figure the light blue and the orange). Their tangents to the first (the orange) at A, B, C, D are labelled 1, 2, 3, 4 and the tangents to the second (the light blue) are labelled 5, 6, 7, 8. Intersections of these tangents are labelled 12, 13, 14, 23, 24, 34 and 56, 57, 58, 67, 68, 78. It is now found that six 6 point conics may be drawn.
Article 229 is When a Conical Quadrilateral produces two 6-point Conics If a quadrilateral ABCD is drawn in a Conic and the tangents are drawn at A, B. C. D then the sides of ABCD and the intersections of the tangents produce two six point conics and a polar line.
Article 230 is A Quadrilateral and the Conics and Points arising ABCD is an arbitrary quadrilateral inscribed in a general conic. The tangents at its vertices form a second quadrilateral PQRS. There emerges the polar line FGTU and 8 points of intersection of ABCD and the tangential quadrilateral PQRS. Many conics result, but two of them in particular meet on the diagonal PR at points X, Y in the figure. X and Y appear to have properties as important as any of the 8 vertices of the quadrilaterals and the four points on the polar line. Not by any means are all the conics that can be drawn featured in this article and a second article is envisaged. What determines why the diagonals PR and QS have such different properties are (as far as we are concerned) open and challenging questions.
Article 231 is A Quadrilateral and resulting Conics Part 2 A Special Case showing the role of the New Points X and Y In Article 230 we investigated properties of a configuration of a quadrilateral ABCD inscribed in an ellipse and the common points of tangents to the ellipse at the vertices A, B, C, D. It turns out that there are two remarkable points X and Y through which many lines may be drawn and also through which many conics pass. The analysis of Part 1 was too complicated to illustrate the results, so in this Article we give a numerical presentation that illustrates the properties of the point X and Y, for which the results all hold in the general case. It seems unlikely the general case can be treated analytically, but possibly a pure approach would succeed as the points are members of a Desargues' involution. Figure 1 above illustrates the line properties of X and Y and Figure 2 below illustrates their Conical properties.
Article 232 is The Coconic Hexagon with Main Diagonals Concurrent ABCDEF is a hexagon inscribed in a conic and is such that its main diagonals AD, BE, CF are concurrent at a point G. Sides AB, BC, … are labelled 1, 2,… . Points such as 14 are then the intersection of AB and DE. Tangents are drawn at A, B, …, so that the tangents at A and B meet at P. Point Q is the intersectionof the tangents at B and C and so on. The tangents at A, B, … are also labelled tA, tB, … and points such as ac are thus the intersections of tA and tC. It is proved that PQRSTU is a conic and that the six points ac, bf, ae, df, ce, bd lie on a conic. It is also proved that 13, 26, 15, 46, 35, 24 lie on a conic.
Article 233 is The Conic and the Lines that are Created (Part 1) A Conic is defined by 5 points, no three of which are collinear, Pairs of lines joining these points and the tangents to the conic at its defining points create 10 lines and these lines generate 15 other key lines and many subsidiary ones.
Article 234 is A Conic generates five 8-point Conics (Part 2) A conic ABCDE with sides 1 = AB, 2 = BC etc. and its tangents a, b, … at the vertices AB, … produce 20 intersections through which five 8-point Conics may be drawn.
Article 235 is The Truth and the whole Truth about a Quadrilateral A quadrilateral is inscribed in a conic and its sides and diagonals and the tangents at its vertices are drawn. A study is made of the lines and conics that pass through the points of intersection. In all a total of eighteen conics emerge three of which pass through eight points and fifteen through six points.
Article 236 is A Nice Conic and a Nasty Circle A triangle ABC and its circumcircle S are given. The line through B parallel to CA and the line through C parallel to AB meet at L. Points M, N are similarly defined by cyclic change. Tangents to S at B and C meet at U, with V, W defined by cyclic change. It is proved that LMNUVW is a conic and the centre of circle UVW is shown to lie on the Euler line of ABC.
Article 237 is How to Construct the Ex-points of a given Point It is shown how knowing the Ex-points of one point it is possible to construct the Ex-points of other points.
Article 238 is When Two Quadrilaterals are in Complete Perspective If two quadrilaterals ABCD and PQRS are such that AP, BQ, CR, DS are concurrent at X then S may be moved into one and only one new position on DX so that the four Desargues' axes coincide.
Article 239 is More on Complete perspective It is shown that if two quadrilaterals ABCD and PQRS are such that AP, BQ, CR, DS are concurrent at a point X (and with the notation that AB^PQ = ab etc.) then if ab, bc, cd are collinear the quadrilaterals are in complete perspective (see Article 238).