Trigonometry/Circles and Triangles/The Nine-point Circle

In any triangle, the following nine points lie on a circle, which is thus called the nine-point circle of that triangle:

  • The mid-points of the three sides;
  • The three points where lines through the vertices perpendicular to the opposite sides meet those sides;
  • The mid-points of the lines between the vertices and the orthocentre.

The radius of the nine-point circle is half that of the circumcircle, and its centre bisects the line between the circumcentre and the orthocentre.

Feuerbach's Theorem

This theorem states that the nine-point circle just touches, without intersecting, the incircle and the three excircles of the triangle. Feuerbach proved this by computing the distances between these circles' centres, and the radii, algebraically.

Last modified on 22 January 2011, at 18:52