Triangle with a 120 degree angle

Here is a beautiful result concerning triangles in which one angle measures 120 degree.


Let ABC be a triangle in which angle A measures 120 degree, and let the internal bisectors of all three angles be drawn; let them meet the opposite sides at P, Q, R (so that AP is the bisector of angle A, BQ is the bisector of angle B, and so on).

Now a beautiful fact emerges: angle QPR is a right angle! 


There are many nice proofs of this fact. Here is a figure showing the result.

Just as beautiful is the fact that the statement has a converse: 


If points P, Q, R are constructed as described above, starting with an arbitrary triangle ABC, and angle QPR is a right angle, then angle A measures 120 degree. 


But this is less easy to prove.


I invite the reader to find both the proofs!