## Quick Reference

Consider a triangle with vertices *A*, *B* and *C*. Let *A*′, *B*′ and *C*′ be the midpoints of the three sides. Let *D*, *E* and *F* be the feet of the perpendiculars from *A*, *B* and *C*, respectively, to the opposite sides. Then *AD*, *BE* and *CF* are concurrent at the orthocentre *H*. Let *P*, *Q* and *R* be the midpoints of *AH*, *BH* and *CH*. The points *A*′, *B*′, *C*′, *D*, *E*, *F*, *P*, *Q* and *R* lie on a circle called the nine-point circle. The centre of this circle lies on the Euler line of the triangle at the midpoint of *OH*, where *O* is the circumcentre. Feuerbach's Theorem states that the nine-point circle touches the incircle and the three excircles of the triangle. This was proved in 1822 by Karl Wilhelm Feuerbach (1800–34).

*Subjects:*
Mathematics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.