The NinePoint CircleThis page requires a javaenabled browser for correct functioning. You can drag the points labelled A, B, and C around with the mouse, and the other points will move accordingly. 
The ninepoint 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 ninepoint circle has been attributed to several different people, including Brianchon and Poncelet, Terquem (who coined the term "ninepoint circle", and (apparently incorrectly, but still commonly done) Euler. Feuerbach's Theorem, published in 1822, states that the incircle and the ninepoint circle are tangent at a point typically called the Feuerbach point. In addition, the ninepoint circle is tangent to the three excircles. 
The radius of the ninepoint circle is half that of the

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