In an isosceles triangle ABC (AB = BC) the heights AM and CK are drawn. Prove that BM = BK.

Since the triangle ABC is isosceles, the angle BAC = BCA.

Since CK and AM are the heights of the triangle ABC, the triangles ACK and AFM are rectangular, in which the hypotenuse of the AC is common, and the angle A = C.

Then the right-angled triangles ACK and ACM are equal in hypotenuse and acute angle, then AK = CM.

Since AB = CB, and AK = CM, then BM = BK, which was required to prove.