Интересная задача о равнобедренном треугольнике с перпендикулярными медианами

Find the angle α at the vertex of an isosceles triangle, knowing that the medians drawn from the endpoints of the base of this triangle are mutually perpendicular. The problem statement is taken from a two-volume mathematics problem book for technical colleges edited by Efimov and Demidovich, published in 1986. The problem is presented in the "Vector Algebra" section, so it is assumed that it should be solved using methods from this branch of mathematics. That's what we'll do! To solve the problem, we'll need the ability to express one vector in terms of another, using, for example, the triangle rule for vector addition. We should also remember the necessary and sufficient condition for two vectors to be orthogonal, as well as the properties of the dot product.