---
tag: D5VS0YFTEZ
kind: definition
name: Simson line
msc: [51M04]
defines: PNRMHP5A0C
uses: [0CTNFGYMM2, 03VZJE6C8C, 0AFNNTRTJE, 0MP9BCF6DP]
sources:
  - url: https://en.wikipedia.org/wiki/Simson_line
---

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane and let $\Gamma$ be its [[circumcircle|03VZJE6C8C]].
Let $P$ be a point of $\Gamma$.
Let $D$, $E$ and $F$ be the feet of the perpendiculars from $P$ to the [[lines|0MP9BCF6DP]] $BC$, $CA$ and $AB$.
The points $D$, $E$ and $F$ are [[collinear|0AFNNTRTJE]], and the *Simson line* of $P$ with respect to $ABC$ is the line that contains them.

## Remarks

The collinearity in the definition is the Simson–Wallace theorem.
Conversely, if the three feet from a point $P$ are collinear, then $P$ lies on $\Gamma$.

Example: let $P = A'$ be the point of $\Gamma$ opposite to $A$.
Since $AA'$ is a diameter, the angles $\angle ABA'$ and $\angle ACA'$ are right angles, so the feet on $AB$ and $CA$ are $B$ and $C$.
The foot on $BC$ lies on the line $BC$, so the Simson line of $A'$ is the line $BC$.

Non-example: let $P$ be the [[circumcentre|0F98FZK14A]] $O$, which is not on $\Gamma$.
The feet of the perpendiculars from $O$ are the [[midpoints|0S706JBZBG]] of the three sides.
These form the medial triangle, so they are not collinear, and no Simson line is defined.