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Twenty-one points on the nine-point circle

The nine points for which the nine-point circle is named are the vertices of three triangles: the medial, orthic, and Euler triangles of a reference triangle ABC. All nine of the vertices lie on the circle, which Dan Pedoe [1] proclaims ‘the first really exciting one to appear in any course on elementary geometry’. An online summary of properties of the circle, which is also called the Euler circle, is given at MathWorld [2].