##
The Simson Line

The sketch below is constructed as follows: For a triangle D*ABC*,
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
*PD*
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 *D*ABC. 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).

Arthur's Home Page

This page was created November 26th, 1998.

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