Question

prove that the length perpendicular from any point on the biscector of an angle to the sides are equal​

Answer 1

The bisector of an angle is constructed by drawing a line through the set of all points that are equidistant to the two sides of the angle. So the distances are equal by definition.

Then, with X and Y the feet of the perpendiculars from D to sides AB and AC respectively, triangle AXD and AYD are congruent.