Interesting problem... I tried solving it through isosceles triangles, and I've proved it doesn't work on equilateral triangles. Can anyone give me some hints towards a geometric proof?
The reason you cannot find a geometric proof is because it is not true. Suppose you have a triangle ABC such that two angle bisectors are perpendicular, say the angle bisectors of A and B. Suppose they intersect at some point D, then we have a right-angled triangle ADB. If in ABC the angle at A is a and the angle at B is b, then in ADB the angle at A is a/2 and the angle at B is b/2. ADB right-angled triangle implies a/2+b/2=90, which implies a+b=180, contradiction.