Question 16
Let $ABCD$ be a rectangle, and let $E$ be any point other than $C$ on side $DC$. Let $H$ be the orthogonal projection of $E$ onto diagonal $AC$, and let $K$ be the orthogonal projection of $C$ onto ray $AE$.
Part a: Prove that $K$ lies on the circumcircle of $ABCD$ and that quadrilateral $CKEH$ is cyclic, that is, can be inscribed in a circle.
Part b: Prove that $\angle CKB+\angle CKH=90^\circ$.
Part c: Prove that $K,H,B$ are collinear if and only if $ABCD$ is a square.
Part a: Prove that $K$ lies on the circumcircle of $ABCD$ and that quadrilateral $CKEH$ is cyclic, that is, can be inscribed in a circle.
Part b: Prove that $\angle CKB+\angle CKH=90^\circ$.
Part c: Prove that $K,H,B$ are collinear if and only if $ABCD$ is a square.