# 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}.

What if applet does not run? |

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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}.

### Inversion - Introduction

- Angle Preservation Property
- Apollonian Circles Theorem
- Archimedes' Twin Circles and a Brother
- Bisectal Circle
- Chain of Inscribed Circles
- Circle Inscribed in a Circular Segment
- Circle Inversion: Reflection in a Circle
- Circle Inversion Tool
- Feuerbach's Theorem: a Proof
- Four Touching Circles
- Hart's Inversor
- Inversion in the Incircle
- Inversion with a Negative Power
- Miquel's Theorem for Circles
- Peaucellier Linkage
- Polar Circle
- Poles and Polars
- Ptolemy by Inversion
- Radical Axis of Circles Inscribed in a Circular Segment
- Steiner's porism
- Stereographic Projection and Inversion
- Tangent Circles and an Isosceles Triangle
- Tangent Circles and an Isosceles Triangle II
- Three Tangents, Three Secants
- Viviani by Inversion
- Simultaneous Diameters in Concurrent Circles
- An Euclidean Construction with Inversion
- Construction and Properties of Mixtilinear Incircles
- Two Quadruplets of Concyclic Points
- Seven and the Eighth Circle Theorem
- Invert Two Circles Into Equal Ones

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Copyright © 1996-2018 Alexander Bogomolny

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