4. A point is on the bisector of an angle if and only if the perpendicular distances from the point to both arms of the angle are the same.

There are two things that must be proved here. The first is that if O is on the bisector of the angle at A, then the perpendicular distance from O to AB is the same as the perpendicular distance from O to AC.

Given: angle OAB = angle OAC,

OB is perpendicular to AB,

OC is perpendicular to AC

To prove: OB = OC

The second thing is that if the perpendicular distance from O is the same to both arms of the angle, that O is on the angle bisector.

Given: OB = OC

OB is perpendicular to AB,

OC is perpendicular to AC

To prove: angle OAB = angle OAC.