Generalized Chapple-Euler Relation

Abstract: We present a new proof of the necessary and sufficient condition for the existence of a triangle that is simultaneously inscribed in a circle and circumscribed about a central conic (an ellipse or a hyperbola). In the limiting case where the foci of the conic coincide, the condition reduces to the classical Chapple-Euler relation. We also prove that the sum of the squared sides lengths of a Poncelet triangle is invariant over a family of Poncelet triangles inscribed in a circle and circumscribed about a central conic if and only if the circle is centered either at the center of the conic or at one of the foci of the conic, among several other properties of such triangles that we derive.