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.

Proof

Copyright © 1996-2009 Alexander Bogomolny

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

• 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
• 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

Copyright © 1996-2009 Alexander Bogomolny