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.