Point M is marked inside isosceles triangle ABC, so AM = BM. Prove that CM is a bisector.

So, in front of us is an isosceles triangle, Connect the vertex C with the point M. Line CM will divide this triangle into two new triangles.
Consider triangles ACM and CMB. They have CM – a common side, AM = BM and the angle between them will also be the same. Thus, we can conclude that the ACM triangle and the CMB triangle are equal by the first sign of equality of the triangles.
This means that the angle of the ACM is equal to the angle of the BCM, and accordingly we can conclude that the CM is a bisector, since it divides the angle C into two equal angles.