- Any point on the bisector of the angle between two lines is equidistant from the two lines. This can be proved using congruent triangles.
- The distance of the point from the line is . The proof of this can be a bit complicated, but there's quite a good one here.
Once you've understood these two facts, the proof of your theorem is very easy. If lies on one of the angle bisectors, then its distance from is , and its distance from is . These two distances are equal, and so