---
tag: PNRMHP5A0C
kind: concept
name: Simson line
msc: [51M04]
sources:
  - url: https://en.wikipedia.org/wiki/Simson_line
same_as:
  wikidata: Q937921
  wikipedia: Simson line
---

## Summary

Let $P$ be a point on the [[circumcircle|03VZJE6C8C]] of a [[triangle|0CTNFGYMM2]].
Its Simson line is the line through the feet of the perpendiculars from $P$ to the three side lines.
These three feet are [[collinear|0AFNNTRTJE]] exactly when $P$ lies on the circumcircle.
The Simson line is a basic object of triangle geometry and appears in many angle chasing arguments.

## Explanations

### L1 human

Pick a point on the circle through the three corners of a triangle.
Drop a perpendicular from it to each of the three sides, extended if needed.
The three feet of these perpendiculars always lie on one straight line.
For example, if the point is a corner $A$, two of the feet are $A$ itself, and the line is the altitude from $A$.