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Two Circles Inscribed in a Parallelogram: What Is This About?
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Copyright © 1996-2008 Alexander Bogomolny
Given a parallelogram ABCD and two circles: one inscribed into angle BAD, the other into angle BCD. Each intersects the diagonal AC twice. For both circles we focus on the points that are closer to the centers of the circles than to the corresponding corners. These points are shown as the yellow dots. Initially, the circles are separated but each can be modified by dragging its center. The configuration can be transformed so as to make the two yellow dots coincide. When this happens, the dot lies on the line of the centers of the two circles, which is to say when the two circles are tangent to or touch one another.
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Why is that so? There are several explanations. One is that when the expanding circles first come into a contact they are bound to share one point only. For, otherwise, it would be possible to shrink them a little still keeping them crossed.
Another explanation invokes the idea of geometric transformations, homothety in particular. The configuration of a pair of side lines (extended), say AB, AD, is homothetic to another pair of the extended side lines CB, CD with respect to any point on the diagonal AC. For an internal point of the diagonal, the homothety coefficient is negative, implying a reflection in that point.
A homothety maps a line onto a parallel line and preserves angles. Therefore, a homothety maps angle bisectors onto each other, and the same holds for the lines joining the centers of the circles with the center of homothety. It follows that the latter are continuation of each other. Which shows that the center line passes through the center of homothety making the circles tangent.
Copyright © 1996-2008 Alexander Bogomolny
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