Poncelet point



In geometry, the Poncelet point of four given points is defined as follows:

Let $A, B, C, D$ be four points in the plane that do not form an orthocentric system and such that no three of them are collinear. The nine-point circles of triangles $△ABC, △BCD, △CDA, △DAB$ meet at one point, the Poncelet point of the points $A, B, C, D$. (If $A, B, C, D$ do form an orthocentric system, then triangles $△ABC, △BCD, △CDA, △DAB$ all share the same nine-point circle, and the Poncelet point is undefined.)

Properties
If $A, B, C, D$ do not lie on a circle, the Poncelet point of $A, B, C, D$ lies on the circumcircle of the pedal triangle of $D$ with respect to triangle $△ABC$ 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 $D$ with respect to triangle $△ABC$, 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 $A, B, C, D$.)

The Poncelet point of $A, B, C, D$ lies on the circle through the intersection of lines $AB$ and $CD$, the intersection of lines $AC$ and $BD$, and the intersection of lines $AD$ and $BC$ (assuming all these intersections exist).

The Poncelet point of $A, B, C, D$ is the center of the unique rectangular hyperbola through $A, B, C, D$.