Tangent Circles in a Parallelogram
What Is It About?
A Mathematical Droodle


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Given a parallelogram ABCD and point T on the diagonal AC. Circles through T inscribed into angles BAD and BCD are tangent at T.

There is more to be said about this configuration.

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Copyright © 1996-2017 Alexander Bogomolny

Let's change the circles while keeping them tangent and inscribed into the angles of parallelogram ABCD. In the applet below me can do that by dragging the center of one of the circle.


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.


What if applet does not run?

The centers move along the angle bisectors of the corresponding angles and the center line remains parallel to itself. This feature implies that the line preserves its length: assuming the two circles are tangent to each other and inscribed into two opposite angles of a parallelogram, the direction and the length of the line joining the centers does not depend on, say, the location of the point of tangency on the diagonal.

The situation here is quite reminiscent of another configuration of two circles and a parallelogram. Our present assertion is a consequence of the properties of that more general configuration.

(The preservation of length property has been observed in a CTK Exchange discussion by Mariano Perez de la Cruz for rectangles and by Bui Quang Tuan for parallelograms.)

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Copyright © 1996-2017 Alexander Bogomolny

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