All triangles are equilateral. Is this true?

In triangle ABC the bisector AF and mediator DF of side BC meet in F. The triangles T_1 and T_2, moreover triangles H_1 and H_2 are equal, thus the sides AB and AC of the triangle ABC are of equal lengths. Repeating for the other sides, e.g AB and BC, w

Information: All triangles are equilateral. Is this true?