Elliptical version of Pythagoras’ Theorem?

If you construct similar shapes on the three edges of a right triangle their areas add up as suggested by the Pythagorean theorem.

In an ellipse, $a^2 = b^2 + c^2$, where $a$ is the semi-major, $b$ is the semi-minor, and $c$ is half the focal distance. In your case, $b=c$, so that $a=\sqrt{2}b$.