High Quality Content by WIKIPEDIA articles! In geometry, the nine-point hyperbola is a hyperbola that passes through the midpoints of the sides of triangle inscribed in a rectangular hyperbola, and that touches six other significant points of the triangle. The nine-point hyperbola was first announced by E.F. Allen in 1941 in the American Mathematical Monthly (48:675–681). Its presentation is easily seen through application of split-complex numbers to an idea Frank Morley had for presenting the nine-point circle with ordinary complex numbers in his book Inversive Geometry (1933). Chapter 15, "The Triangle", starts with the fact that any triangle is similar to one with vertices on the unit circle. Then the orthocenter is the sum of these complex numbers, and the center of the nine-point circle is half that number, and the radius is 1/2. Allen's title is "On a triangle inscribed in a rectangular hyperbola" and he takes the hyperbola zz* = 1 in the split-complex plane.

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