Inversions preserve angles

The following script shows that inversion in the circle preserves angles.   An angle at P can be thought of as the sum (or difference) of two angles, both made with respect to a ray through O.  Thus, we need only check angles like the one in the following sketch.  The angle is at P, and is created by the ray OP and the line PQ.  The image of the ray OP is itself, and the image of the line PQ is the circle through O.  The image of P is P', and we want to check that the angle the tangent to the circle at P' makes with the ray OP is the same as the angle at P.  That is, we want to check that the angle RPP' is the same as the angle RP'P.  The following does this.  Change the location of P by moving P, and change the angle at P by moving Q.  You can change the circle we are inverting in by moving O and S.

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This java applet was produced using JavaSketchpad.  The following is the maker's beta on this product:

This is a prototype of JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum Press, Inc. All rights reserved. Portions of this work were funded by the National Science Foundation (awards DMI 9561674 & 9623018).

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This page was created November 26th, 1998.