Prove if the centers of the incircle and the circumcircle lie on the median of the triangle, then this triangle is slave.

The center of the circumscribed circle near the triangle is the point of intersection of the mid-perpendiculars, then the OH segment is perpendicular to AC.

The center of the circle inscribed in the triangle is the point of intersection of the bisectors, then ВН is the bisector of the angle ABC.

The centers of the circle O, O1, by condition, lie on the median BH, and since it is also the bisector and height, the triangle ABC is isosceles or equilateral, which was required to prove.