If do not lie on a circle, the Poncelet point of lies on the
circumcircle of the
pedal triangle of with respect to triangle and lies on the other analogous circles. (If they do lie on a circle, then those pedal triangles will be lines; namely, the
Simson line of with respect to triangle , and the other analogous Simson lines. In that case, those lines still concur at the Poncelet point, which will also be the
anticenter of the cyclic quadrilateral whose vertices are .) The Poncelet point of lies on the circle through the intersection of lines and , the intersection of lines and , and the intersection of lines and (assuming all these intersections exist). The Poncelet point of is the center of the unique
rectangular hyperbola through . In the case that form an orthocentric system and the Poncelet point is undefined, there is instead a family of rectangular hyperbolas through , and the common nine-point circle is the locus of their centers. ==References==