KöMaL 2021 C1652

The shorter leg of each of two right-angled triangles has unit length. In each triangle, the right-angled vertex is at a unit distance from a point dividing the hypotenuse in a $2:1$ ratio: in one case it is the point closer to the right-angled vertex, and in the other case it is the point farther away. Prove that it is possible to select three out of the non-unit sides of the two triangles such that the three lengths form a right-angled triangle.