Question 96
Let $ABC$ be a triangle, let $O$ and $\gamma$ be its circumcenter and circumcircle, respectively, and let $P$ and $Q$ be distinct points interior to $\gamma$ such that $O, P$ and $Q$ are not collinear. Reflect $O$ in the midpoint of the segment $PQ$ to obtain $R$, then reflect $R$ in the center of the nine-point circle of the triangle $ABC$ to obtain $S$. The circle through $P$ and $Q$, and orthogonal to $\gamma$, crosses the rays $OP$ and $OQ$, emanating from $O$, again at $P'$ and $Q'$, respectively. Let the lines $PQ'$ and $QP'$ cross at $T$. Prove that, if $P$ and $Q$ are isogonally conjugate with respect to the triangle $ABC$, then so are $S$ and $T$.
E. D. Camier, England
The points $U$ and $V$ are isogonally conjugate with respect to the triangle $ABC$ if $\angle(AB, AU) = \angle(AV, AC)$ and $\angle(BC, BU) = \angle(BV, BA)$, in which case $\angle(CA, CU) = \angle(CV, CB)$ as well.
E. D. Camier, England
The points $U$ and $V$ are isogonally conjugate with respect to the triangle $ABC$ if $\angle(AB, AU) = \angle(AV, AC)$ and $\angle(BC, BU) = \angle(BV, BA)$, in which case $\angle(CA, CU) = \angle(CV, CB)$ as well.