Miquel's Theorem for Circles

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Miquel's Theorem for Circles

Let there be two triples of points A₀, A₁, A₂ and B₀, B₁, B₂. The applet below illustrates the following statement: if circles A₀B₁B₂, B₀A₁B₂, B₀B₁A₂ concur in a point V different from any of the given six, then the same holds for the circles B₀A₁A₂, A₀B₁A₂, A₀A₁B₂.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Proof

The statement is a consequence of Miquel's theorem and basic properties of inversion. Indeed, suppose circles A₀B₁B₂, B₀A₁B₂, B₀B₁A₂ meet in point V. An inversion with center V converts the three circles into three straight lines that form a triangle B'₀B'₁B'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'₀A'₁A'₂, A'₀B'₁A'₂, A'₀A'₁B'₂ meet in a point, the Miquel point of the configuration. The latter is the inversive image of the common point of circles B₀A₁A₂, A₀B₁A₂, A₀A₁B₂.

Inversion - Introduction

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