Question 24
Let $ABC$ be an acute triangle with $\widehat A=45^\circ$, and let $P$ be the foot of the altitude from $B$. Draw the circle with centre $P$ through $C$, meeting $AC$ again at $X$ and the altitude $PB$ at $Y$. Let $r,s$ be the lines through $P,X$, respectively, perpendicular to $AY$, and let $L,K$ be their intersections with $AB$. Prove that $L$ is the midpoint of $KB$.