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%% MAIN DOCUMENT
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\begin{document}
\begin{center}
\Large {\bf BMOC Mentoring Scheme}\\
\large Advanced Level -- Sheet 4\\
January 2002\\
\end{center}
\vskip 40pt
\begin{enumerate}
\item
\begin{enumerate}
\item ({\bf The Angle Bisector Theorem}) Let ABC be a triangle, L a point on
the interior of the side BC.  Show that AL is the internal bisector of
the angle CAB if and only if
$${BL \over LC} = {AB \over AC}$$
What is the corresponding theorem for external angle bisectors?
\item In quadrilateral ABCD, the internal bisectors of angles A and C
meet on BD; prove that the internal bisectors of angles B and D meet
on AC.
\end{enumerate}
\item
\begin{enumerate}
\item ({\bf Ceva's theorem}) Let ABC be a triangle, and suppose L,M,N are
points on sides BC, CA, AB respectively.  Prove that AL, BM, CN being
concurrent is equivalent to
$${BL \over LC}\cdot {CM \over MA}\cdot {AN \over NB} = 1$$
and to
$${\sin BAL \over \sin LAC}\cdot {\sin CBM \over \sin MBA}\cdot {\sin
ACN \over \sin NCB} = 1$$
\item Let ABC be a triangle and D,E,F be points on sides BC, CA, AB
respectively such that AD, BE, CF are concurrent; if D', E', F' are
the reflections of D,E,F in the midpoints of BC, CA, AB respectively,
prove that AD', BE', CF' are concurrent.  Prove also that the
reflection of AD in the internal bisector of angle A, the reflection
of BE in the internal bisector of angle B, and the reflection of CF in
the internal bisector of angle C are concurrent.
\end{enumerate}
\item
\begin{enumerate}
\item If $\alpha$, $\beta$, $\gamma$, $\delta$ are complex numbers,
show that
$$(\alpha-\beta)(\gamma - \delta) + (\alpha-\delta)(\beta-\gamma) =
(\alpha - \gamma)(\beta - \delta)$$
\item ({\bf The Ptolemy--Euler Inequality}) Deduce that if A,B,C,D are
any four points in the plane then
$$AB\cdot CD + BC\cdot DA \ge AC\cdot BD$$
When does equality occur?
\item If P lies on the arc AB of the circle circumscribed about a
regular pentagon ABCDE, show that
$$PC + PE = PA+PB+PD$$
\end{enumerate}
\item
\begin{enumerate}
\item ({\bf Miquel's Theorem}) If the points L,M,N lie on sides
BC,CA,AB respectively of triangle ABC (possibly extended), prove that
the circles AMN, BNL, CLM are concurrent at a point P.
\item Prove that $\measuredangle CPB = \measuredangle CAB +
\measuredangle NLM$.
\item ({\bf The Simson Line}) Prove that the feet of the
perpendiculars to the sides of a triangle from a point are colinear if
and only if the point lies on the circumcircle of the triangle.
\item Given four lines in the plane, prove that the circumcircles of
the four triangles they form are concurrent.
\end{enumerate}
\item Triangle ABC has incentre I, and L, M, N are the midpoints of
BC, CA, AB respectively.  Prove that I lies inside triangle LMN.
\item Let ABC be a triangle.  Suppose X,Y are points on side BC such
that $\measuredangle BAX = \measuredangle YAC$.  If the incircle of
triangle ABX touches BX at L and the incircle of triangle ACY touches
CY at M, prove that
$${1 \over BL} + {1 \over LX} = {1 \over CM} + {1 \over MY}$$
\item The triangle ABC is right--angled at A.  A line through the
midpoint D of BC meets AB at X and AC at Y (AB and AC being extended
as necessary).  The point P is taken on this line so that PD and XY
have the same midpoint M.  The perpendicular from P to BC meets BC at
T.  Prove that AM bisects angle TAD.
\item In a scalene triangle ABC, D is the foot of the perpendicular
from A to BC, E and F are the midpoints of AD and BC respectively, and
G is the foot of the perpendicular from B to AF.  Prove that EF is the
tangent at F to the circle GFC.
\item The acute--angled triangle ABC has circumcentre O and
orthocentre H.  The altitude BH meets circle ABC again at P and OP
meets CA at Q.  The altitude CH meets the circle again at R and OR
meets AB at S.  Prove that the lines PQ, QH, HS, SR touch a circle.
What happens if $A=45^\circ$?
\end{enumerate}
\end{document}