# Miquel's Theorem for Circles

Let there be two triples of points A_{0}, A_{1}, A_{2} and B_{0}, B_{1}, B_{2}. The applet below illustrates the following statement: if circles A_{0}B_{1}B_{2}, B_{0}A_{1}B_{2}, B_{0}B_{1}A_{2} concur in a point V different from any of the given six, then the same holds for the circles B_{0}A_{1}A_{2}, A_{0}B_{1}A_{2}, A_{0}A_{1}B_{2}.

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Copyright © 1996-2018 Alexander Bogomolny

The statement is a consequence of Miquel's theorem and basic properties of inversion. Indeed, suppose circles A_{0}B_{1}B_{2}, B_{0}A_{1}B_{2}, B_{0}B_{1}A_{2} meet in point V. An inversion with center V converts the three circles into three straight lines that form a triangle B'_{0}B'_{1}B'_{2} with images B' of points B and points A' - the images of points A - on the sides of this triangle. According to Miquel's theorem, the circumcircles of triangles B'_{0}A'_{1}A'_{2}, A'_{0}B'_{1}A'_{2}, A'_{0}A'_{1}B'_{2} meet in a point, the Miquel point of the configuration. The latter is the inversive image of the common point of circles B_{0}A_{1}A_{2}, A_{0}B_{1}A_{2}, A_{0}A_{1}B_{2}.

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Copyright © 1996-2018 Alexander Bogomolny