[Math] bisectors of an angle in a triangle intersect at a single point – proof verification

Question

Let´s consider a general triangle ABC. Let´s draw two angle bisectors from vertices A and B.

It is obvious that these two angle bisectors intersect at a single point X. Since X lies on the angle bisector from A its distance from side b is the same as its distance from side c. As X also lies on the angle bisector from B, its distance from side c is the same as its distance from side a.

By transitivity the distance from the point X to the side b has to be the same as the distance from X to the side a. Therefore the angle bisector from C (which is the set of points equidistant from b and a) has to go through X.

It seems as a very simple argument. Is it correct?

Answer 1

This is correct subject to the comments.