Simson’s line

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

In the picture, the line passing through $U,V,W$ is a Simson line for $\mathrm{△}ABC$.

An interesting result form the realm of analytic geometry 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