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.
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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 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
- 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
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
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