# The Nine-Point Circle

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The nine-point circle, discovered by Feuerbach in about 1820, contains the midpoints of the three sides of the triangle and the feet of the three altitudes. Its center is the midpoint of the segment joining the orthocenter and the circumcenter of the triangle, and consequently lies on the Euler line.

The discovery of the nine-point circle has been attributed to several different people, including Brianchon and Poncelet, Terquem (who coined the term "nine-point circle", and (apparently incorrectly, but still commonly done) Euler.

Feuerbach's Theorem, published in 1822, states that the incircle and the nine-point circle are tangent at a point typically called the Feuerbach point. In addition, the nine-point circle is tangent to the three excircles.

The radius of the nine-point circle is half that of the circumcircle, and midpoint of any line segment between the orthocenter and the circumcircle lies on the nine-point circle.

Java images created using Cinderella by Scott Sutherland on March 13, 2004.