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\begin{document}
\begin{center}
{\large\bf On isogonal conjugacy, Miquel points,
(Newton-)Gauss lines, etc.}
{\bf N.Beluhov, A.Zaslavsky, P.Kozhevnikov}
\end{center}
\section{Introductory problems}
\zd {\bf Isogonal conjugacy.} Given a triangle $ABC$ and a point
$P$.
\pp Prove that lines symmetric to $AP$, $BP$, $CP$
in the bisectors of corresponding angles are concurrent or parallel.
The common point $P'$ of these lines is called {\it isogonal conjugate} to $P$
with respect to $ABC$.
\pp Prove that $P'$ is a point at infinity (i.e. three corresponding lines
are parallel) iff
$P$ lies on the circumcircle of $ABC$.
\pp Determine the image isogonal conjugacy of a circle passing through
two of three points $A$, $B$, $C$.
{\footnotesize \pp\footnote{Here and further we footnotesize
the statements that not used in the proofs of results from section 2.}
Prove that all projections of $P$ and $P'$ to the sidelines of $ABC$ are concyclic.
Reformulate the statement above for the case when $P'$ is a point at infinity.
\pp For two pairs $X, X'$ and $Y, Y'$ of isogonal conjugate points,
prove that $XY\cap X'Y'$ and $XY'\cap X'Y$ are isogonal conjugates.}
% задача об изогональном сопряжении четырехсторонника
\vspace{1mm}
Given a quadrilateral $ABCD$ and a point
$P$.
\pp Suppose that three of four lines symmetric to $AP$, $BP$, $CP$, $DP$
in the bisectors of corresponding angles are concurrent.
Prove that all four lines are concurrent.
\pp Prove that for a point $P$ there exists
an isogonal conjugate $P'$ iff
projections of $P$ to the sidelines of $ABCD$ are concyclic (if $P'$ exists, then
all projections of $P$ and $P'$ to the sidelines of $ABCD$ are concyclic).
%$^N$ если проекции лежат
%на одной прямой, то изогонально сопряженная точка --- бесконечно удаленная).
\vspace{1mm}
{\footnotesize A conic is said to be inscribed to a polygon if it touches
all the sidelines of this polygon.
\pp Prove that foci of a conic inscribed to a triangle are isogonal conjugates.
\pp Prove that focus of a parabola inscribed to a triangle
lies on its circumcircle.}
\vspace{1mm}
\zd {\bf Miquel point.} Given a quadrilateral $ABCD$.
Let $E=AB\cap CD$, $F= AD\cap BC$.
\pp Prove that (in notation of the previous problem)
circumcircles of triangles $ABF$, $CDF$,
$ADE$, $CDE$ have a common point $M$
(Miquel point for a quadruple of lines $AB$, $BC$, $CD$, $DA$).
\pp Prove that $M$ is a center of spiral similarity
that takes segment $BE$ to $FD$ (or $DE$ to $FB$, etc.)
{\footnotesize \pp Two bugs $B$ and $C$ move, each at a constant speed,
along two lines intersecting at $A$. Prove that all the circles
$ABC$ have a common point, and $^*$ all the lines $BC$ touch fixed parabola.
\pp (IMO2005) Let $ABCD$ be a convex quadrilateral with sides
$BC$ and $AD$ equal in length and not parallel. Let $E$ and
$F$ be interior points of the sides $BC$ and $AD$ such that
$BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines
$BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$
meet at $R$. Consider all triangles $PQR$
as $E$ and $F$ vary. Prove that
the circumcircles of these triangles have a common point other than $P$.
\pp Establish a connection between Miquel point and inscribed conics.
\pp Prove that the projections of Miquel point to the sidelines of a quadrilateral
lie on a line perpendicular to Gauss line. Establish a connection between this line and a parabola inscribed
to the quadrilateral.}
\vspace{1mm}
\zd {\bf Gauss line.} Given a quadrilateral $ABCD$. Let
$E = AB\cap CD$, $F=AD\cap BC$.
\pp Prove that the midpoints of the segments $AC$, $BD$, $EF$ lie on a line
(that is called Gauss line of $ABCD$, or Gauss line of quadruple of lines $AB$, $BC$, $CD$, $DA$).
\pp Prove that the center of the circle passing through the projections of
a pair of isogonal conjugates
lies on Gauss line.
\pp Prove that Miquel point is isogonal conjugate
to the infinite point of Gauss line.
{\footnotesize \pp Prove that centers of conics inscribed to a quadrilateral
lie on Gauss line.
\pp (All-Russian Olympiad 2009)
Let $A_1$ and $C_1$ be points on the sides $AB$ and $BC$ of parallelogram $ABCD$.
Let $P = AC_1\cap CA_1$.
Circumcircles of triangles~$AA_1P$ and $CC_1P$ meet for the second time at point $Q$
lying inside triangle~$ACD$. Prove that $\angle PDA=\angle QBA$.}
\section{Three Miquels for a Quartet.}
In this section we use the following notation.
Let $A$, $B$, $C$, $D$ be four points such that no three of them are collinear.
Let $X$ be Miquel point for the quadruple of lines $AB$, $AC$, $BD$, $CD$, let $Y$
be Miquel point for the quadruple of lines $AB$, $AD$, $BC$, $CD$, let $Z$
be Miquel point for the quadruple of lines $BC$, $AC$, $BD$, $AD$.
We set $P_X=AD\cap BC$, $P_Y=AC\cap BD$, $P_Z=AB\cap CD$.
Let $K_X$ and $L_X$ be midpoints of the segments $BC$ and $AD$ respectively,
similarly, let $K_Y$, $L_Y$ be midpoints of
$AC$, $BD$, let $K_Z$, $L_Z$ be midpoints of $AB$, $CD$.
Let $\Gamma_X = K_XL_X$, $\Gamma_Y = K_YL_Y$, $\Gamma_Z=K_ZL_Z$ be Gauss lines
for the corresponding quadruples of lines.
\vspace{1mm}
%\zd Докажите, что прямые, проходящие через $A$, $B$, $C$ и
%параллельные $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, пересекаются в
%одной точке или параллельны.
\zd Prove that $AX$, $BY$, $CZ$ have a common point
$D'$, or parallel. Similarly define
$A'$, $B'$, $C'$.
%perspective centers for triangle $XYZ$ and triangles
%$DCB$, $CDA$, $BAD$.
\vspace{1mm}
\zd Prove that $A'$, $B'$, $C'$, $D'$ are isogonal conjugates to
$A$, $B$, $C$, $D$ with respect to triangle $XYZ$.
\vspace{1mm}
\zd Prove that $X$, $Y$, $Z$ are Miquel points for quadruples of lines joining
$A'$, $B'$, $C'$, $D'$.
\vspace{1mm}
\zd Probe that lines $AA'$, $BB'$, $CC'$, $DD'$ are parallel.
\vspace{1mm}
\zd Prove that $AD$, $A'D'$, $YZ$ are concurrent
(find other analogous intersections).
%здесь для решения можно использовать задачу об изогональном сопряжении четырехсторонника
\vspace{1mm}
\zd \pp Prove that points $X$, $Z$, $P_Y$, $K_Y$, $K_Y$ lie on a certain circle
$\omega_Y$. Similarly define circles $\omega_X$, $\omega_Z$.
\pp Prove that $\omega_X$, $\omega_Y$, $\omega_Z$ have a common point $T$.
\pp Prove that $XP_X$, $YP_Y$, $ZP_Z$ meet at $T$.
\eject
\section{Quartets for three Miquels.}
Let $XYZ$ be a triangle. Define a transformation $\psi_X$ as the symmetry
in the bisector of angle $X$ followed by the inversion
with center $X$ and radius $R = \sqrt{XY\cdot XZ}$.
Similarly define transformations $\psi_Y$, $\psi_Z$.
\zd Prove that
\pp $\psi_X(Y)=Z$, $\psi_X(Z)=Y$;
\pp $\psi_X^2$ is the identity transformation;
\pp Product $\psi_Z \psi_Y \psi_X$ is the identity transformation.
Let $D$ be an arbitrary point, let $A=\psi_X(D)$, $B=\psi_Y(D)$,
$C=\psi_Z(D)$.
\zd Prove that $\triangle XDZ\sim\triangle XYA$ and $\triangle
XDY\sim\triangle XZA$.
\zd Prove that each of the transformations $\psi_X$, $\psi_Y$,
$\psi_Z$ takes the 4-element set $\{A, B, C, D\}$ to itself.
A 4-element set of points $\{A, B, C, D\}$ defined as above is said to be
a {\it quartet}. From the previous problem it follows that all the plane except $X$, $Y$, $Z$
is partitioned into quartets.
\zd Prove that four isogonal conjugates to points of a quartet is a quartet.
\zd Find all the quartets containing
\pp the incenter $I$ of triangle $XYZ$;
\pp the circumcenter $O$ of triangle $XYZ$.
\pp Find the invariant points for $\psi_Z$, and corresponding quartets.
\zd \pp Prove that $X$ is Miquel point for the quadruple of lines $AB$, $AC$,
$BD$, $CD$.
\pp Formulate similar statements for $Y$, $Z$.
\pp Prove the converse: if $X$, $Y$, $Z$ are Miquel points defined by
$A$, $B$, $C$, $D$, then $A$, $B$, $C$,
$D$ us a quartet (for $X$, $Y$, $Z$).
%\zd Докажите, что прямые $AX$, $BY$, $CZ$ пересекаются в одной
%точке.
\zd Prove that each of transformations $\psi_X$, $\psi_Y$,
$\psi_Z$ commutes with the isogonal conjugacy with respect to $XYZ$.
{\footnotesize \zd Suppose $A$, $B$, $C$, $D$ be a quartet with respect to
$XYZ$, let $A'$, $B'$, $C'$, $D'$ be isogonal conjugates to $A$, $B$, $C$, $D$ respectively.
Consider four conics having pairs of foci $A$
and $A'$, $B$ and $B'$, $C$ and $C'$, $D$ and $D'$.
\pp Prove that these conics are homothetic to each other.
\pp Prove that midpoints of six segments joining centers of these conics
lie on a certain conic that is homothetic to them and passing through $X$, $Y$, $Z$.
\zd Let $M$, $N$ be a pair of isogonal conjugates with respect to triangle $ABC$
lying inside $ABC$. It appears that $AM\cdot AN\cdot BC=BM\cdot BN\cdot
AC=CM\cdot CN\cdot AB=k$.
\pp Prove that the midpoint of $MN$ is the gravity center of $A$, $B$, $C$.
\pp Find $k$ in terms of side lengths of $ABC$.}
\eject
\section{Additional problems.}
\zd \pp Let $A,B,C,D$ be a quartet, $A',B',C',D'$ be conjugated quartet; let $P_X$ be intersection point of $AD$ and $BC$, $P_Y$~--- of $AC$ and $BD$, $P_Z$~--- of $AB$ and $CD$. Points $Q_X,Q_Y,Q_Z$ are defined similarly by points $A',B',C',D'$. Prove that lines $P_XQ_X,P_YQ_Y,P_ZQ_Z$ are concurent in the point, which lie on the circumcircle of triangle $XYZ$ (notations as above).
\pp In previous notations prove that lines $P_XQ_Y,P_YQ_X$ and $XY$ concur.
\pp Let $Z'$ be the point obtained in b). Prove that line $ZZ'$ is parallel to $AA',BB',CC',DD'$.
\pp Let $D_1,D_1'$ and $D_2,D_2'$ be two pairs of isogonally conjugated points such that $D_1D_1'\parallel D_2D_2'$. Prove that lines $A_1A_2,B_1B_2,C_1C_2,D_1D_2$ concur ($A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are quartets).
\zd Given points $A,B,C,D$. It is known that triangle $XYZ$ is perspective to each of triangles $ABC,BCD,CDA,DAB$ (with indicated order of vertices). Points $D',A',B',C'$ are respective centers of perspective. Prove that lines $AA',BB',CC',DD'$ concur.
\end{document}