Firstly, it is proved that the triangle ABE and ACD are congruent: AB=AC, with a common angle and a right angle.

So there is angle ABE= angle ACD. The two base angles of an isosceles triangle are equal. So CBO angle = BCO angle can be proved to be isosceles.

AD=AE can be obtained from congruence. The male side AO has a right angle, which proves that the triangle is congruent. It can be proved that AO is the bisector of angle BAC. Add the triangle ABC isosceles, and you can get the name AO vertical BC.