The Simson Line
The sketch below is constructed as follows: For a triangle DABC,
we construct the circumcircle and choose a point P on this circle.
The point D is the point on the (extended) side BC such that
is perpendicular to BC. The points E and
F are similarly
defined. (Don't take my word for this -- check that these are infact
perpendicular.) A priori, there is no reason to expect that
the three points D, E and F are colinear, but the
sketch infact suggests that they are.
Move the point P around the circle. Note that the three points
remain colinear (and are still the bases of perpendiculars through P).
The triangle, too, can be changed -- move any one of the vertices around.
This sketch is a demonstration of the following theorem:
Theorem: Let P be a point on the circumcircle of an
arbitrary triangle DABC. Let D,
E and F be the bases of perpendiculars through P to the (extended) sides
BC, AC and AB respectively. Then D, E and F are colinear. This
line is called the Simson line.
The java applet above was produced by JavaSketchpad. The following
is the maker's beta:
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.