Question 55
In a right triangle $ABC$ with $\angle ACB = 90^\circ$ let $D$ be the foot of the altitude from $C$. Let $E$ and $F$ be the reflections of $D$ with respect to $AC$ and $BC$, respectively. Let $O_1$ and $O_2$ be the circumcenters of the triangles $ECB$ and $FCA$, respectively. Prove that
$$
2|O_1O_2| = |AB|.
$$
$$
2|O_1O_2| = |AB|.
$$