# 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 $\triangle ABC$.

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 SimsonsLine 2013-03-22 12:24:34 2013-03-22 12:24:34 drini (3) drini (3) 17 drini (3) Theorem msc 51-00 Circumcircle Triangle