In isosceles triangle ABC (AB = BC), bisectors AM and CK are drawn, which intersect at point O.

In isosceles triangle ABC (AB = BC), bisectors AM and CK are drawn, which intersect at point O. Prove that triangle AOK = triangle COM.

Let us prove the equality of triangles ACK and ACM.

In triangles, the AC side is common and the angles at the base are equal, since the angle ACM = CAM as angles at the base of an isosceles triangle, angle CAM = ACK as half the angles at the base, then triangles ACK and ACM are equal in side and adjacent angles, then the angle AKC = CMA, segment AK = CM.

In triangles AOK and COM, the sides AK and CM are equal, the angle KAO = COM, the angle AKO = CMO, then the triangles AOK and COM are equal in side and adjacent angles, which was required to prove.