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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*.
####

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).

Arthur's Home Page

This page was created November 26th, 1998.

URL: http://www.nevada.edu/~baragar/geom/angle.htm