Poncelet point

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

Let A, B, C, and D be four points in the plane. The nine-point circles of triangles ABC, BCD, CDA, DAB meet at one point, the Poncelet point of the points A, B, C, and D.

The Poncelet point may not exist. Necessary and sufficient conditions for this to happen are that the quadrilateral ABCD is orthocentric, or two of the points A, B, C, D coincide.