# nine-point circle

The nine point circle also known as the Euler’s circle or the Feuerbach circle^{} is the circle that passes through the feet of perpendiculars^{} from the vertices $A,B$ and $C$ of a triangle^{} $\mathrm{\u25b3}ABC.$

Some of the properties of this circle are:

Property 1 :
This circle also passes through the midpoints^{} of the sides $AB,BC$ and $CA$ of $\mathrm{\u25b3}ABC.$ This was shown by Euler.

Property 2 :
Feuerbach showed that this circle also passes through the midpoints of the line segments^{} $AH,BH$ and $CH$ which are drawn from the vertices of $\mathrm{\u25b3}ABC$ to its orthocenter^{} $H.$

These three triples of points make nine in all, giving the circle its name.

Property 3 :
The radius of the nine-point cirlce is $R/2,$ where $R$ is the circumradius^{} (radius of the circumcircle).

Property 4 :
The center of the nine-point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on the Euler line^{}.

Property 5 :
All triangles inscribed^{} in a given circle and having the same orthocenter, have the same nine-point circle.

Title | nine-point circle |
---|---|

Canonical name | NinepointCircle |

Date of creation | 2013-03-22 13:11:20 |

Last modified on | 2013-03-22 13:11:20 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 6 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 51-00 |

Synonym | Euler circle |

Synonym | Feuerbach circle |

Synonym | nine point circle |