In the isosceles triangle ABC with the base of the AC, the medians AE and CD are drawn. Prove that triangle DOE

In the isosceles triangle ABC with the base of the AC, the medians AE and CD are drawn. Prove that triangle DOE and triangle AOE are isosceles, where O is the intersection point of AE and CD

Since, by condition, AE and CD are the medians of the triangle, points D and E divide the sides in half. AD = BD, CE = BE, and since the triangle is isosceles, then AD = CE.

The angles BAC and BCA at the base of the AC are equal, then the quadrilateral ADEC is an isosceles trapezoid, and the segments CD and AE of its diagonals, which are equal and at the point of intersection are divided into equal segments. ОD = ОЕ, ОА = OC, and therefore the triangles DOE and AOE are isosceles, as required.