---
tag: 8K5FT58EK5
kind: concept
name: Orthocentric system
msc: [51M04]
sources:
  - url: https://en.wikipedia.org/wiki/Orthocentric_system
same_as:
  wikidata: Q897296
  wikipedia: Orthocentric system
---

## Summary

An orthocentric system is a set of four points of the plane.
In it, one point is the [[orthocentre|0WB619CA0V]] of the triangle formed by the other three.
Then each of the four points is the orthocentre of the triangle formed by the other three.
The four triangles share the same [[nine-point circle|033H8MQRC3]], and their circumcircles all have the same radius.

## Explanations

### L1 human

Take a triangle that has no right angle and mark its orthocentre, where the altitudes meet.
Now you have four points.
Pick any three of them: their orthocentre is the fourth point.
So the four points play symmetric roles, and none of them is special.
For example, $(0, 2)$, $(-1, 0)$, $(1, 0)$ and $(0, \frac{1}{2})$ form such a system.