Question 28
Let $I$, $O$, $J_A$ be the centers of the incircle, circumcircle, and excircle corresponding to the side $BC$ of a triangle $ABC$ with $AC \gt AB$, respectively. Let $r$, $R$, $r_a$ be the radii of these circles, respectively. Let the incircle touch the side $BC$ at $D$ and $E$ be a point on the line segment $BD$ different from the endpoints, such that the area of the triangle $IEJ_A$ is twice the area of the triangle $IEO$. Prove that
$$
ED = AC - AB \iff R = 2r + r_a.
$$
$$
ED = AC - AB \iff R = 2r + r_a.
$$