solve the angle bisector problem

My geometry teacher at St. Ignatius College Prep (Fall 1969) Mr. O’Mara promised to add 5 points to our grade when we submitted a proof for the following problem: Given that two angle bisectors are of equal length, prove that the triangle is issoceles. Mr. O’Mara is dead but the problem is still rattling around in my head. I can prove the rererse but can’t seem to figure out how to go the other way. Sometimes I think the problem is intractable, other times I think it requires some fancy compass and arc drawing work.