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 $\mathrm{△}ABC.$

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 $\mathrm{△}ABC.$ 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 $\mathrm{△}ABC$ 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