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\begin{document}
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\begin{flushright}
{\bf A.Zaslavsky, A.Akopyan,\\ P.Kozhevnikov, A.Zaslavsky}
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\begin{center}
{\bf\large Brocard points}
\end{center}
\vskip 12pt
\section{Brocard points in triangles}
\ \par\vskip-\baselineskip
\zd Let a triangle $ABC$ be given. Prove that there exists a unique point $P$, such that $\angle PAB=\angle PBC=\angle PCA=\phi_1$, and a unique point $Q$,
such that $\angle QBA=\angle QCB=\angle QAC=\phi_2$.
{\bf Definition 1.} Points $P$ and $Q$ are called the {\it Brocard points} of triangle $ABC$.
\zd
\pp Prove that $\phi_1=\phi_2=\phi$.
\pp Find $\phi$ as a function of the angles of $ABC$.
{\bf Definition 2.} Angle $\phi$ is called the {\it Brocard angle} of triangle $ABC$.
\zd Prove that the projections of Brocard points to the sidelines of $ABC$ are concyclic. (This is true for any pair of isogonally conjugated points).
\zd Let $O$ be the circumcircle of $ABC$.
\pp Prove that $OP=OQ$.
\pp Prove that $\angle POQ=2\phi$.
{\bf Definition 3.} The reflections of the medians of a triangle in its correspondent bisectors are called the {\it symmedians}. Three symmedians concur in point $L$, which is called the {\it Lemoine point} of the triangle.
\zd Prove that $P$ and $Q$ lie on the circle with diameter $OL$.
\zd (K.Knop) Consider two triangles: one of them is formed by the circumcenters of triangles $PAB$, $PBC$, $PCA$; the second one is formed by the circumcenters of triangles $QAB$, $QBC$, $QCA$. Prove that these triangles are
\pp similar to $ABC$;
\pp equal.
\pp Find the center and the angle of the rotation transforming one of these triangles to the second one.
\zd Let $C'$ be a point of segment $AB$, such that $AB$ is the external bisector of angle $PC'Q$. Prove that $CC'$ is the symmedian of $ABC$. (I.e. there exists an ellipse with foci $P$ and $Q$ touching the sides of the triangle in the bases of its symmedians).
\zd Let $T_1$, $T_2$ be points of line $OL$, such that $\angle LPT_1=\angle LPT_2=60^{\circ}$. Prove that the projections of each of these points to the sidelines of $ABC$ form a regular triangle (these points are called the {\it Apollonius points}).
\eject
\section{Brocard points in quadrilaterals}
\ \par\vskip-\baselineskip
\zd Let $ABCD$ be a convex broken line. Prove that there exists a unique point $P$, such that $\angle PAB=\angle PBC=\angle PCD=\phi$.
{\bf Definition 4.} We will call $P$ and $\phi$ the {\it Brocard point} and the {\it Brocard angle} of broken line $ABCD$. We will denote them as $P(ABCD)$ and $\phi(ABCD)$.
\zd Find $\phi(ABCD)$ as a function of the lengths of segments $AB$, $BC$, $CA$ and the angles between them.
\zd Prove that $\phi(ABCD)=\phi(DCBA)$ iff $A$, $B$, $C$, $D$ are concyclic.
Now we will consider only cyclic polygons.
\zd Let $P_1=P(ABCD)$, $P_2=P(BCDA)$, $P_3=P(CDAB)$, $P_4=P(DABC)$.
Prove that $P_1P_2P_3P_4$ is a cyclic quadrilateral.
\zd Let $Q_1=P(DCBA)$, $Q_2=P(ADCB)$, $Q_3=P(BADC)$, $Q_4=P(CBDA)$.
Prove that $P_1P_2/Q_1Q_2=BC/CD$, $P_2P_3/Q_2Q_3=CD/DA$ etc.
\zd (D.Belev) Let $M_1$, $M_2$ be points on lines $AD$, $AB$ respectively such that $BM_1\parallel CD$, $CM_2\parallel DA$.
\pp Prove that the circumcircles of triangles $BAM_1$ and $BCM_2$ meet in $P_1$.
\pp Define the similar construction for $P_i, i=2,\ldots, 4$, $Q_i, i=1,\ldots,4$.
\zd (D.Belev) Prove that lines $CP_1$, $DP_2$, $AP_3$, $BP_4$ concur, and lines $BQ_1$, $CQ_2$, $DQ_3$, $AQ_4$ concur.
\zd (D.Belev) Denote the points obtained in the previous problem as $P_0$, $Q_0$.
\pp Prove that $S_{P_1P_2P_0}=S_{Q_1Q_2Q_0}$
\pp Prove that the areas of $P_1P_2P_3P_4$ and $Q_1Q_2Q_3Q_4$ are equal.
\zd Prove that $\phi(ABCD)=\phi(BCDA)$ iff $AB\cdot CD=AD\cdot BC$.
{\bf Definition 5.} A cyclic quadrilateral with equal products of opposite sides is called {\it harmonic}. From the last problem we obtain that in the harmonic quadrilateral there exist points $P$ and $Q$, such that $\angle PAB=\angle PBC=\angle PCD=\angle PDA=\angle QDC=\angle QCB=
\angle QBA=\angle QAD=\phi$. We will call $P$, $Q$ and $\phi$ the {\it Brocard points} and the {\it Brocard angle} of quadrilateral $ABCD$.
\zd Prove that each of the following conditions is true iff $ABCD$ is harmonic.
\pp The tangents to the circumcircle in $A$ and $C$ meet on $BD$.
\pp $BD$ is a symmedian of $ABC$.
\pp The distances from the common point $L$ of the diagonals to the sides are proportional to these sides.
\pp There exists an inversion transforming $A$, $B$, $C$, $D$ to the vertices of a square.
\pp There exists a central projection transforming $ABCD$ and its circumcircle to a square and a circle.
\zd Find the Brocard angle of a harmonic quadrilateral as a function of its angles.
\zd Prove that $OP=OQ$ and $\angle POQ=2\phi$.
\zd Prove that $P$ and $Q$ lie on the circle with diameter $OL$.
\eject
\includegraphics{brockard.eps}
\eject
\section{Brocard points in polygons}
\ \par\vskip-\baselineskip
\zd Let a circle, a point $P$ inside it and an angle $\phi$ be given. For an arbitrary point $X_0$ on the circle construct a point $X_1$, such that the oriented angle $PX_0X_1$ is equal to $\phi$. Similarly for $X_1$ construct $X_2$ etc. Prove that if $X_n=X_0$, then this is true for any other initial point.
\zd Find the closure condition in the previous problem.
Remind that all considered polygons are cyclic.
{\bf Definition 6.} We will call a polygon $A_1\ldots A_n$ a {\it Brocard} polygon if there exists a point $P$, such that $\angle PA_1A_2=
\angle PA_2A_3=\cdots\angle PA_nA_1=\phi$.
\zd Prove that in a Brocard polygon there exists a point $Q$ such that $\angle QA_1A_n=\angle QA_nA_{n-1}=\cdots\angle QA_2A_1=\phi$.
{\bf Definition 7.} We will call $P$, $Q$ and $\phi$ the {\it Brocard points} and the {\it Brocard angle} of $A_1\ldots A_n$.
\zd Prove that each of the following conditions is true iff $A_1\ldots A_n$ is the Brocard polygon.
\pp There exists a point $L$, such that the distances from it to the sides of the polygon are proportional to these sides.
\pp The symmedians of triangles $A_1A_2A_3, A_2A_3A_4,\ldots, A_nA_1A_2$ from $A_2, A_3,\ldots, A_1$ concur.
\pp The common points of lines $A_1A_3, A_2A_4,\ldots, A_nA_2$ with the tangents to the circumcircle in $A_2, A_3,\ldots, A_1$ respectively are collinear.
\pp There exists an inversion transforming $A_1,\ldots, A_n$ to the vertices of a regular triangle.
\pp There exists a central projection transforming the polygon and its circumcircle to a regular polygon and a circle.
\zd Prove that the Brocard points lie on the circle with diameter $OL$ and $\angle POL=\angle QOL=\phi$.
\zd
\pp Prove that there exist two points $T_1$, $T_2$ such that the inversion with the center in any of them transforms $A_1,\ldots,A_n$ to the vertices of a regular triangle.
\pp Prove that $T_1$, $T_2$ lie on $OL$ and $\angle T_1PL=
\angle T_2PL=\frac{\pi}n$.
\zd Find the Brocard angle as a function of $OL/R$.
\end{document}