# Four Touching Circles III

The applet below helps verify the validity of the theorem about a chain of four circles tangent in pairs. There are four contact points. These may or may not lie in a circle. They do if the number of external tangencies at these points is even; they do not, otherwise.

All circles in the applet are draggable. Each circle is defined by its center and by a point on its border. Both points are also draggable and can be used to modify the size of the circles. To see the points check the "draggable points" checkbox.

You can invert the diagram (check "inversion") in a circle that can be dragged and resized. Letting its center overlap one of the contact points illustrates the proof of the theorem discussed on a separate page.

Try dragging the circles around as to make them touch in pairs. The applet will supply a circle if there are at least three contact points.

What if applet does not run? |

Here's one case with one pair of circles touching internally:

and another with three internal tangencies:

### Relevant pages:

- The Four Touching Circles Problem
- The Four Touching Circles Problem II
- The Four Touching Circles Problem III
- The Four Touching Circles Problem IV

### 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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