##  [Nine-Point Circle Theorem](/nine-point-circle-theorem) 

  ##  [Nine-Point Circle Theorem](https://puremath.quantumdictionary.io/nine-point-circle-theorem-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The theorem that the following nine points of a triangle are concyclic: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from the orthocenter to each vertex; the resulting circle is called the nine-point circle, its center N is the midpoint of the orthocenter–circumcenter segment and its radius equals half the circumradius.