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Nine point circle

In geometry, the nine point circle is a circle that can be constructed for any given triangle. It is named so because it passes through nine significant points, with six of them lying on the triangle itself: the midpoints of the three sides, the feet of the altitudess, and the midpoints of the portion of altitude between the vertices and the orthocenter. It is also known as Feuerbach's circle, Euler's circle, Terquem's circle, six-points circle, twelve-points circle, n-point circle, medioscribed circle, mid circle or circum-midcircle.

In the diagram above, the points are:

D, E, F - the midpoints of the three sides
G, H, I - the feet of the altitudes
J, K, L - the points on each altitude midway between the vertex and the orthocentre (labelled S)

The nine point circle is tangent externally to the three excircles and tangent internally to the incircle of the triangle, a theorem discovered by Karl Wilhelm Feuerbach in 1822 in the form:
... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle...
The following image illustrates this theorem:

The point at which the incircle and the nine point circle touch is often called the Feuerbach point.

Feuerbach was not the first to discover the circle. At a slightly earlier date, Charles Brianchon and Jean-Victor Poncelet had stated and proven the same theorem. Soon after Feuerbach, mathematician Olry Terquem also proved what Feuerbach did and added the three points that are the midpoints of the altitude between the vertices and the orthocenter. Terquem was the first to use the name nine point circle (as he was the first to associate nine special points with the circle).

Other facts of interest:

The center of the nine point circle (the nine point center) lies on the triangle's Euler line, at the midpoint between the triangle's orthocenter and circumcenter.
  • If an orthocentric system of four points is given, any three of them define a triangle, and these four triangles all have the same nine point circle.
  • The centers of the incircle and excircles of a triangle form an orthocentric system. The nine point circle created for that orthocentric system is the circumcircle of the original triangle.
  • See also

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