nine-point circle

The nine point circle also known as the Euler’s circle or the Feuerbach circleMathworldPlanetmath is the circle that passes through the feet of perpendicularsPlanetmathPlanetmath from the vertices A,B and C of a triangleMathworldPlanetmath ABC.

Some of the properties of this circle are:

Property 1 : This circle also passes through the midpointsMathworldPlanetmathPlanetmathPlanetmath of the sides AB,BC and CA of ABC. This was shown by Euler.

Property 2 : Feuerbach showed that this circle also passes through the midpoints of the line segmentsMathworldPlanetmath AH,BH and CH which are drawn from the vertices of ABC to its orthocenterMathworldPlanetmath 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 circumradiusMathworldPlanetmath (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 lineMathworldPlanetmath.

Property 5 : All triangles inscribedMathworldPlanetmath 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