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## Nine point circleIn geometry, thenine 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 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 point at which the incircle and the nine point circle touch is often called the 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 radius of the nine point circle is half the length of the radius of the circumcircle of the triangle.
- The nine point circle bisects any line from the orthocenter to a point on the circumcircle.
nine point center) lies on the triangle's Euler line, at the midpoint between the triangle's orthocenter and circumcenter.
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