Question 71
Let $ABC$ be an acute triangle with circumcenter $O$. A circle $k$ has the following properties:
(1) Its center $K$ lies in the interior of side $BC$.
(2) It is tangent to $AB$ at $B'$ and to $AC$ at $C'$.
(3) $O$ lies on the shorter arc $B'C'$ of $k$.
Prove that the circumcircle of $ABC$ and $k$ intersect at two distinct points.
(1) Its center $K$ lies in the interior of side $BC$.
(2) It is tangent to $AB$ at $B'$ and to $AC$ at $C'$.
(3) $O$ lies on the shorter arc $B'C'$ of $k$.
Prove that the circumcircle of $ABC$ and $k$ intersect at two distinct points.