Question 173
Let $ABC$ be an acute, non-isosceles triangle which circumcircle $(O)$. Take $M$ on segment $BC$ and $J$ on $(O)$ such that $\angle BAJ = \angle CAM$. Take a point $D$ on the segment $AM$ and denote $O_1, O_2$ as the circumcenters of triangles $ABD$ and $ACD$. The line $O_1O_2$ meets $BC$ at $T$ and $G$ is the second intersection of circumcircles of triangles $TBO_1$ and $TCO_2$. Prove that $OA$, $JA$ respectively divide the segments $O_1O_2$, $BC$ by the same ratio and $A$, $G$, $O$, $J$ are concyclic.
Problem:
Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$ and circumcenter $O$. Denote $D$, $E$ as midpoints of segments $AB$, $AC$. Take $M$, $N$ on $BC$ such that $MB = BC = CN$ ($B$ is between $M$, $C$ and $C$ is between $N$, $B$). Denote $P$, $Q$ as the projections of $H$ onto the lines $BE$, $CD$. The circumcircles of triangle $ABN$, $ACM$ respectively meets $AQ$, $AP$ again at $Y$, $X$. Suppose that the line $XY$ cuts $BC$ at $K$. Prove that $AK$ is perpendicular to $OH$.
Problem:
Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$ and circumcenter $O$. Denote $D$, $E$ as midpoints of segments $AB$, $AC$. Take $M$, $N$ on $BC$ such that $MB = BC = CN$ ($B$ is between $M$, $C$ and $C$ is between $N$, $B$). Denote $P$, $Q$ as the projections of $H$ onto the lines $BE$, $CD$. The circumcircles of triangle $ABN$, $ACM$ respectively meets $AQ$, $AP$ again at $Y$, $X$. Suppose that the line $XY$ cuts $BC$ at $K$. Prove that $AK$ is perpendicular to $OH$.