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\begin{document}
\title{Inversion (and Transformation Geometry)}%
\author{Peter McNamara}%
\date{April 14 2006}
\maketitle
\section{Problems}
\begin{enumerate}
\item Two common tangents of two intersecting circles meet at a
point $A$. Let $B$ be a point of intersection of the two circles,
and $C$ and $D$ be the points in which one of the tangents touches
the circles. Prove that the line $AB$ is tangent to the circle
passing through $B$, $C$ and $D$.
\item Suppose that $\Gamma$ is a cirle, internally tangent to two
other intersecting circles $\Gamma_1$ and $\Gamma_2$ at points $X$
and $Y$ respectively. $\Gamma_1$ and $\Gamma_2$ intersect at $A$
and $B$. $C$ is a point of intersection of $AB$ and $\Gamma$, and
let the lines $CX$ and $CY$ meet $\Gamma_1$ and $\Gamma_2$
respectively at $D$ and $E$ (different from $X$ and $Y$). Prove
that $DE$ is tangent to both $\Gamma_1$ and $\Gamma_2$.
\item Let $ABC$ be a triangle, right angled at $A$, $D$ the foot
of the altitude from $A$, and $E$, $F$ the incentres of triangles
$ABD$ and $ACD$ respectively. $EF$ intersects $AB$ and $AC$ in $K$
and $L$. Show $AK=AL$.
\item Let $ABCD$ be a convex quadrilateral. On the segments $AB$,
$BC$, $CD$, $DA$ are chosen points $M$, $N$, $P$, $Q$
respectively, such that $$AQ=DP=CN=BM.$$ Show that if $MNPQ$ is a
square, then $ABCD$ is a square.
\item Prove Feuerbach's theorem; that the nine point circle is
tangent to the incircle and the three excircles.
\item In the plane we are given two circles intersecting at $X$
and $Y$. Prove that there exist four points with the following
property: For every circle touching the given circles at $A$ and
$B$, and meeting the line $XY$ at $C$ and $D$, each of the lines
$AC$, $AD$, $BC$, $BD$ passes through one of these four points.
\item Let $C$ be a circle inside another circle $\Gamma$. Let
$T_0$ be a circle tangent externally to $C$ and internally to
$\Gamma$. For each $k\geq1$, let $T_k$ be the circle, distinct
from $T_{k-2}$ that is externally tangent to $C$ and $T_{k-1}$ and
internally tangent to $\Gamma$. Prove that if $T_n=T_0$ for one
particular such circle $T_0$, then $T_n=T_0$ for all possible
positions of $T_0$.
\item Let $A_1$ be the centre of the square inscribed in an acute
triangle $ABC$ with two vertices on $BC$ and one vertex each on
$AB$ and $CA$. Similarly define $B_1$ and $C_1$. Prove that the
lines $AA_1$, $BB_1$ and $CC_1$ are concurrent.
\item (APMO 1998 Q4) Let $ABC$ be a triangle and $D$ the foot of
the altitude from $A$. Let $E$ and $F$ be on a line passing
through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is
perpendicular to $CF$, and $E$ and $F$ are different from $D$. Let
$M$ and $N$ be the midpoints of the line segments $BC$ and $EF$
respectively. Prove that $AN$ is perpendicular to $NM$.
\item Let $ABC$ be an acute triangle with circumcentre $O$ and
circumradius $R$. Let $AO$ meet the circumcircle of $OBC$ again at
$D$, $BO$ meet the circumcircle of $OCA$ again at $E$, $CO$ meet
the circumcircle of $OAB$ again at $F$. Show that $OD\cdot OE\cdot
OF\geq 8R^3$.
\item Circle $\Gamma_1$ is internally tangent to circle $\Gamma_2$ at $D$. At a
point $B$ on $\Gamma_1$, different from $D$, the tangent to
$\Gamma_1$ is drawn and intersects $\Gamma_2$ at $A$ and $C$.
Prove that $BD$ bisects $\angle ADC$.
\item Let $ABC$ be a triangle, let $\Gamma$ be its incircle and
$\Gamma_a$, $\Gamma_b$ and $\Gamma_c$ be three circles orthogonal
to $\Gamma$ passing through $(B,C)$, $(A,C)$ and $(A,B)$,
respectively. The circles $\Gamma_a$ and $\Gamma_b$ meet again in
$C'$; and in the same way we obtain the points $B'$ and $A'$.
Prove that the radius of the circumcircle of $A'B'C'$ is half the
radius of $\Gamma$.
\item $ABCD$ is a quadrilateral inscribed in the circle with
diameter $AD$. Let $O$ be the midpoint of $AD$, $M$ the
intersection of $BC$ and $AD$, and $K$ the second intersection
point of the circumcircles of triangles $BAO$ and $CDO$. Prove
that $\angle MKO=90^{\circ}$.
%\item (2000 IMO Q6) Let $AH_1$, $BH_2$ $CH_3$ be the altitudes of
%an acute angled triangle $ABC$. The incircle of the triangle $ABC$
%touches the sides $BC$, $CA$ and $AB$ at $T_1$, $T_2$ and $T_3$
%respectively. Let the lines $\ell_1$, $\ell_2$ and $\ell_3$ be the
%reflections of the lines $H_2H_3$, $H_3H_1$ and $H_1H_2$ in the
%lines $T_2T_3$, $T_3T_1$ and $T_1T_2$ respectively. Prove that
%$\ell_1$, $\ell_2$ and $\ell_3$ determine a triangle whose
%vertices lie on the incircle of the triangle $ABC$.
%\item On the sides $AB$, $BC$, $CD$, $DA$ of a quadrilateral
%$ABCD$, we construct alternately to the outside and inside,
%regular triangles with vertices $Y$, $W$, $X$, $Z$ respectively.
%Show that $YWXZ$ is a parallelogram.
\item Two circles intersect in points $A$, $B$. A line $\ell$ that
contains the point $A$ intersects the circles again in the points
$C$, $D$. Let $M$, $N$ be the midpoints of the arcs $BC$ and $BD$
respectively, which do not contain the point $A$, and let $K$ be
the midpoint of the segment $CD$. Show that $\angle
MKN=90^{\circ}$.
\item (1990 USAMO Q5) An acute-angled triangle $ABC$ is given in
the plane. The circle with diameter $\, AB \,$ intersects altitude
$\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$,
and the circle with diameter $\, AC \,$ intersects altitude $\,
BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that
the points $\, M, N, P, Q \,$ lie on a common circle.
\item Let $AD, BE, CF$ be altitudes of an acute triangle $ABC$
with $AB>AC$. Line $EF$ meets $BC$ at $P$, and the line through
$D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$
respectively. Let $N$ be any point on the side $BC$ such that
$\angle NQP +\angle NRP<180\,^{\circ}$. Prove that $BN>CN$.
\end{enumerate}
\end{document}