Question 191
An acute scalene triangle $ABC$ with $\angle A = 45^\circ$ is inscribed in a circle $(O)$. Let $AD$ be the altitude and $H$ be the orthocenter of $\triangle ABC$. Points $R$ and $S$ different from $A$ are taken on the rays $AB$ and $AC$ respectively so that $CA = CR$ and $BA = BS$. Points $X, Y$ and $K$ are the midpoints of the segments $BS, CR$ and $AH$, respectively. Point $T$ lies on the line $BC$ such that $AO$ is perpendicular to $TO$, and $I$ is the circumcenter of triangle $DXY$. Prove that
$$
XY \perp OK \text{ and } TH \perp IK.
$$
$$
XY \perp OK \text{ and } TH \perp IK.
$$