7. Triangle $A B C$ is the right angled triangle with the vertex $C$ at the right angle. Let $P$ be the point of reflection of $C$ about $A B$. It is known that $P$ and two midpoints of two sides of $A B C$ lie on a line. Find the angles of the triangle.
Solution. Let $C$ be the vertex at the right angle and $C H$ be the altitude. The point $P$ symmetric to $C$ relative to $A B$ is on line $C H$ (continuation of the altitude) and $C H=P H$. Let $M$ and $N$ be the midpoints of $A C$ and $A B$, respectively. Let us also draw the median $C N$. We have $C N=B N$ (indeed, $N$ is the centre of the circumscribed circle about $A B C$ ). Triangles $\triangle B C H$ and $\triangle N H P$ are equal, hence $P N=a$ and thus $C N=P N=a$. We also know that $B N=C N=a$, hence $\triangle B C N$ is equilateral.
Answer: $30^{\circ}, 60^{\circ}, 90^{\circ}$.