Let AM be the median of a right-angled triangle ABC drawn from the vertex of the right angle A, and P and Q

Let AM be the median of a right-angled triangle ABC drawn from the vertex of the right angle A, and P and Q be the tangency points of the circle inscribed in the triangle ABM, with its sides AB and BM, respectively. It is known that PQ is parallel to AM. Find the angles of triangle ABC.

Consider the triangle ABM:

РQ parallel to AM, it means that the triangle ABM is equilateral (only in an equilateral triangle, the segment connecting the points of tangency is parallel to the base).

In equilateral triangles, all angles are 60 °, so the angle B is 60 °.

Angle A is equal to 90 ° (by condition), which means that angle C = 180 ° – (90 ° + 60 °) = 180 ° – 150 ° = 30 °.

Answer: angle A = 90 °, angle B = 60 °, angle C = 30 °.