In an isosceles triangle ABC with base AC, points M and N are the midpoints of sides AB

In an isosceles triangle ABC with base AC, points M and N are the midpoints of sides AB and BC, respectively. Prove the equality of triangles AMC and CNA.

Since the triangle is isosceles, then AB = BC, and since points M and N are the midpoints of the sides, then AM = BM = BN = CN.

In triangles AMC and CNA, the AC side is common. Angle NCA = MAC, since the angles at the base are equal in an isosceles triangle.

Then the triangles AMC and CNA are equal in two sides and the angle between them – the first criterion for the equality of triangles, which was required to prove.