For the 2nd problem

from the 2 right triangles you are told exist, using the pythagorean theorem

$x^2 + h^2 = a^2$

$y^2 + h^2 = b^2$

now you are told that $h^2 = x y$

substituting this in we get

$x^2 + xy = a^2$

$y^2 + xy = b^2$

adding these two equations we get

$x^2 + y^2 + 2 x y = a^2 + b^2$

$(x+y)^2 = a^2 + b^2$

and looking at triangle ACB we see that $x+y$ is the hypotenuse of the triangle with legs $a$ and $b$ and as

$(x+y)^2 = a^2 + b^2$ ACB must be a right triangle.