Question 155
Let $\triangle ABC$ be a right triangle with the right angle at $C$ and let $D$ be a point on the segment $BC$. Denote the circumcircle of the triangle $ABD$ by $\mathcal{K}$. Let $E$ be a point on $\mathcal{K}$, such that the chord $DE$ is perpendicular to $AB$. Prove that the triangle $AEB$ is isosceles with the apex at $B$ if and only if $CA$ is tangent to $\mathcal{K}$.