Triangle ABC is isosceles, AC is base, BD is median. Prove that triangle ABD and CBD are right-angled.

Since, according to the condition, the triangle ABC is isosceles with the base of the AC, then AB = BC, the angle BAC = BC.

The BH segment is the median of the ABC triangle, then AD = CD.

In triangles ABD and BCD AD = CD, AB = BC, angle BAD = BCD, then triangles ABD and BCD are equal in two sides and the angle between them, which means the angle ADB = CDB.

The angle ADB is adjacent to the angle CDB, the sum of which is 180, and since these angles are equal, then the angle ADB = CDB = 90, which means that the triangles ABD and CBD are rectangular, which was required to be proved.