Poncelet point
In geometry, the Poncelet point of four given points is defined as follows:
Let Template:Mvar 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 Template:Math meet at one point, the Poncelet point of the points Template:Mvar. (If Template:Mvar do form an orthocentric system, then triangles Template:Math all share the same nine-point circle, and the Poncelet point is undefined.)
Properties
If Template:Mvar do not lie on a circle, the Poncelet point of Template:Mvar lies on the circumcircle of the pedal triangle of Template:Mvar with respect to triangle Template:Math 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 Template:Mvar with respect to triangle Template:Math, 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 Template:Mvar.)
The Poncelet point of Template:Mvar lies on the circle through the intersection of lines Template:Mvar and Template:Mvar, the intersection of lines Template:Mvar and Template:Mvar, and the intersection of lines Template:Mvar and Template:Mvar (assuming all these intersections exist).
The Poncelet point of Template:Mvar is the center of the unique rectangular hyperbola through Template:Mvar.
References
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