Generalized Chapple--Euler Relation

We provide a new proof of the necessary and sufficient condition for a triangle to be circumscribed about a central conic (ellipse or hyperbola), expressed in terms of the circumradius and the distances from the circumcenter to the foci. If the inscribed conic is an ellipse, in the limiting case, where the foci coincide, the condition reduces to the classical Chapple--Euler relation. We also prove that the sum of the squares of the sides of a triangle in a family inscribed in a circle and circumscribed about a central conic remains invariant throughout the family if and only if the center of the circle coincides either with the center of the conic or with one of its foci. Using Blaschke products of degree three and affine transformations, we characterize all central $3$-Poncelet pairs with constant triangle area. We prove that the associated family of triangles has constant area if and only if the two conics are homothetic ellipses. We also prove two observations of Reznik concerning the invariance of the total area of the power circles of Poncelet triangles by relating them to Apollonius's identity on medians and establish a more general result valid for both ellipses and hyperbolas. Finally, we propose two conjectures on area invariance of power-circles.