Tangency Point of Two Circles: What is this about?
A Mathematical Droodle

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(Points A and B move horizontally. Points C and D move vertically.)

Explanation

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Copyright © 1996-2018 Alexander Bogomolny

Assume two circles tangent to AB at A and B are also tangent to each other. What is the locus of their common point T when their radii change?

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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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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What if applet does not run?

The locus is a circle with center at the midpoint of AB and the radius half its length. To see that draw the common tangent to the two circles at T to its intersection with AB at O. Then, since two tangents from a point to a circle are equal,

OA = OT = OB.

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Copyright © 1996-2018 Alexander Bogomolny