QD=QE, OQ is the same side, QD is perpendicular to OA, QE is perpendicular to OB, and the right triangle ODQ is the congruent right triangle OEQ, which proves that Q is on the bisector.

The intersection q is the perpendicular of AB, QA=QB, and the triangle QAB is an isosceles triangle. According to the vertical theorem of isosceles triangle, point Q is on AB perpendicular bisector.