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 $\gamma^2=2\alpha\beta$?
$AB$ is the diameter of the red circle. If $P$ belongs to the red circle, $\angle APB=90^\circ$ (semi-circle). So by the Pythagorean theorem, $AB^2=AP^2+PB^2\implies (\alpha+\gamma+\beta)^2=(\alpha+\gamma)^2+(\beta+\gamma)^2\implies\gamma^2=2\alpha\beta.$
The converse of the Pythagorean theorem takes care of the "$\Leftarrow$" direction.