Poncelet point
In geometry, the Poncelet point of four given points is defined as follows:
Let be four points in the plane that do not form an orthocentric system such that no three of them are collinear. The nine-point circles of triangles meet at one point, the Poncelet point of the points.
Properties
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.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 .
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.