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.