Inversive distance and Poncelet pairs

The eccentricity of the inner ellipse in a 3-Poncelet pair with the unit circle coincides with the pseudo-hyperbolic distance between its foci. Based on Barbilian’s metrization procedure, which employs Apollonian circles to express the pseudo-hyperbolic distance within the unit disk, a new proof of this fact is obtained together with a geometric construction of the corresponding ellipse. Two pairs of Apollonian circles, classical and dual, arise as natural geometric and metric guides, their inversive distance and intersection angle being directly related to the ellipse’s eccentricity. The methods are elementary and establish new connections among inversive geometry, hyperbolic geometry, and Poncelet configurations.