This problem is a converse to one that I found in the journal "Crux Mathematicorum with Mathematical Mayhem" published by the Canadian Mathematical Society".

Let I and J be the incircle and excircle of triangle ABC in the interior of angle ABC. I and J touch side AC at points D and E respectively. Line BE intersects I at points F and G with F between B and G. Prove that DF is a diameter of I.