proof of Simson’s line

Given a ABC with a point P on its circumcircleMathworldPlanetmath (other than A,B,C), we will prove that the feet of the perpendicularsMathworldPlanetmathPlanetmathPlanetmathPlanetmath drawn from P to the sides AB,BC,CA (or their prolongations) are collinearMathworldPlanetmath.

Since PW is perpendicular to BW and PU is perpendicular to BU the point P lies on the circumcircle of BUW.

By similarMathworldPlanetmathPlanetmath arguments, P also lies on the circumcircle of AWV and CUV.

This implies that PUBW , PUCV and PVWA are all cyclic quadrilateralsMathworldPlanetmath.

Since PUBW is a cyclic quadrilateral,




Also CPAB is a cyclic quadrilateral, therefore


(opposite angles in a cyclic quarilateral are supplementaryPlanetmathPlanetmath).

From these two, we get


Subracting CPW, we have


Now, since PVWA is a cyclic quadrilateral,


also, since UPVC is a cyclic quadrilateral,


Combining these two results with the previous one, we have


This implies that the points U,V,W are collinear.

Title proof of Simson’s line
Canonical name ProofOfSimsonsLine
Date of creation 2013-03-22 13:08:26
Last modified on 2013-03-22 13:08:26
Owner giri (919)
Last modified by giri (919)
Numerical id 9
Author giri (919)
Entry type Proof
Classification msc 51-00