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.

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

and another with three internal tangencies:

Inversion

• 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 Tool
• Feuerbach's Theorem: a Proof
• Four Touching Circles
• Four Touching Circles III
• Hart's Inversor
• Inversion in the Incircle
• Inversion: Reflection in a Circle
• Inversion with a Negative Power
• Peaucellier Linkage
• Polar Circle
• Poles and Polars
• Radical Axis of Circles Inscribed in a Circular Segment
• Steiner's porism
• Three Tangents, Three Secants

Copyright © 1996-2010 Alexander Bogomolny