Question

Find the angles of the triangle, if the midpoint of one of its bisectors coincides with the midpoint of the segment connecting feet of the height and the median drawn from the other two vertices.

Answer 1

all are 60°

Explanation:

The midpoints of the angle bisector and the foot of the median will lie on the midsegment of the triangle. In order for the foot of the altitude to lie on that same midsegment, the altitude must be a median.

If the angle bisector intersects the midpoint of the midsegment, it, too, is a median. The angle bisector will be a median only if the triangle is isosceles with the bisector's angle being the a.pex. The altitude will be a median only if the altitude's vertex is the a.pex of an isosceles triangle. Hence, the triangle must be equilateral, and all angles are 60°.