If I am not wrong it is an easy task

The first case will give us the clue for the whole solution.

Since the sum of sides a+b must be equal to a given quantity, that means that vertex C is on an ellipse with foci A an B, on the other hand, since C is the vertex of the right angle, the hypotenuse c of our triangle must be on a semi-circumference of diameter AB what yield that point C is the point of tangent between both, so sides a,b and the altitude lengths are equal, obviously side c must be equal to the sum of sides a+b

Established the above, for the second and third cases this ratio among sides and hypotenuse it must be preserved.

Algebraically It can be also proved, but I believe it is less intuitive