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 and such that no three of them are collinear. The nine-point circles of triangles meet at one point, the Poncelet point of the points . (If do form an orthocentric system, then triangles all share the same nine-point circle, and the Poncelet point is undefined.)
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 .