In this picture,
$ABC$ is an equilateral triangle, whereas $ABP$ is a rectangle triangle. Let $P$ be inside the equilateral triangle, and $\alpha,\beta,\gamma$ three segments such that they sum up to the side of $ABC$ (and also to the hypotenuse of $ABP$, by construction).
Is it true that $P$ belongs to the red circle if and only $\gamma^2=2\alpha\beta$?