Prove that the bisector of an interior angle of a triangle a

Prove that the bisector of an interior angle of a triangle and the bisectors of the remote exterior angles are concurrent.

Solution

Let\'s consider that a bisector (internal or external) of the angle made by two distinct lines 1,2 meeting at some point P is (part of) the locus of points Q for which d(Q,1)=d(Q,2)

If some point Q belongs to the external angle bisector through B and to the external angle bisector through C, it fulfills

d(Q,BA)=d(Q,BC)=d(Q,CA)

so by d(Q,BA)=d(Q,CA), it belongs to the internal angle bisector through A