Question 137
Given a circle $\omega$ with diameter $A B$ and a point $P$ inside the segment $A B$, let $M$ be the midpoint of $P B$. Let $r, s$ be two parallel lines passing through $M, P$ respectively, not coinciding with the line $A B$ nor perpendicular to it. Let $H$ be the orthogonal projection of $A$ onto $s$ and let $K$ be the intersection point (distinct from $A$) of $\omega$ and the line $A H$. Finally, let $X, Y$ be the intersections of $r$ with $\omega$, where $X$ is on the opposite side of $H$ with respect to $A B$.
a. Prove that triangle $H Y K$ is isosceles.
b. Prove that $B X H Y$ is a parallelogram.
a. Prove that triangle $H Y K$ is isosceles.
b. Prove that $B X H Y$ is a parallelogram.