Simson’s line

Let ABC a triangle and P a point on its circumcircleMathworldPlanetmath (other than A,B,C). Then the feet of the perpendicularsMathworldPlanetmathPlanetmathPlanetmath drawn from P to the sides AB,BC,CA (or their prolongations) are collinearMathworldPlanetmath.

In the picture, the line passing through U,V,W is a Simson lineMathworldPlanetmath for ABC.

An interesting result form the realm of analytic geometryMathworldPlanetmath states that the envelope formed by Simson’s lines when P varies is a circular hypocycloid of three points.

Title Simson’s line
Canonical name SimsonsLine
Date of creation 2013-03-22 12:24:34
Last modified on 2013-03-22 12:24:34
Owner drini (3)
Last modified by drini (3)
Numerical id 17
Author drini (3)
Entry type Theorem
Classification msc 51-00
Related topic Circumcircle
Related topic Triangle