The AC and BM segments intersect and the intersection point is divided in half.

The AC and ВM segments intersect and the intersection point is divided in half. Prove that triangle ABC is equal to triangle CMA.

Given:
Segments AC and BM intersect at point O,
AO = OС,
ВO = OM.
Prove that triangle ABC is equal to triangle CMA.
Evidence:
1) Consider the triangle AOM and BOС. They have AO = OС, ВO = OM, angle AOM = angle ВOС, as they are vertical, then on two sides and the angle between them, the triangles AOM = ВOС. So AM = BC;
2) Consider the triangle AOB and MOС. They have AO = OС, BO = OM, angle AOB = angle MOC, since they are vertical, then on two sides and the angle between them the triangles AOB = MOC. So AB = MC;
3) Triangle ABC = CMA on three sides, since AM = BC, AB = MC and BM are common. Proven.