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. . . . . . In this page, the concepts of orthogonal circles and inversion geometry will be presented.
should describe orthogonal circles and the inversion circle, and such things as the angle preservation property etc.
From Doctor Math, Orthogonal Circles
From Mathworld, Orthogonal Circles, Inversion
From Cut-the-knot, Inversion: Reflection in a Circle and Inversion circle
Paul Kunkel's Inversion Geometry
Triangle centers, incircle, excircles, and the nine-point circle -- the nine-point circle is tangent to the incircle and the three excircles by Feuerbach's Theorem, a fact that is proved on cut-the-knot using inversion circles.
The webmaster and author of this Math Help site is Graeme McRae.