60^o Breeds 90^o: What is this about?
A Mathematical Droodle

Explanation

The applet is supposed to suggest the following problem:

Extend AB beyond A to, say, X. Since ∠BAC = 120^o, ∠CAX = 60^o. Also ∠CAF = 60^o, so that AC is the external angle bisector at A of ΔABF. The external angle bisector at A intersects the (interior) angle bisector at B in an excenter of ΔABF. It follows that G is an excenter of that triangle. Another external angle bisector passes through G: that of the exterior angle at F. Therefore, FG is the bisector of ∠AFC. Similarly, FH is the bisector of ∠AFB. Since the two angles are supplementary, so that the angle between FG and FH is indeed 90^o.

References

1. D. Fomin, A. Kirichenko, Leningrad Mathematical Olympiads 1987-1991, MathPro Press, 1994, pp. 38, 146-7