Based on BC of isosceles triangle ABC, points M and K are taken so that BM = CK Prove that AM = AK.

By connecting point A with points M and K, we have three triangles. Two of which will be equal.

ΔАВМ = ΔАСК – by the first sign of equality of triangles (If two sides and the angle between them of one triangle are equal respectively to the two sides and the angle between them of another triangle, then such triangles are equal.)

In our triangles AB = AC (ΔABС is isosceles, and in an isosceles triangle the sides are equal), BM = KC (by condition), <B = <C (in an isosceles triangle, the angles at the base are equal).

The equality of triangles ABM and AСK implies the equality of the sides AM = AK, which was required to prove.