Last modified on 10 April 2011, at 17:32

Trigonometry/Circles and Triangles/Orthocentric points

As previously noted, the altitudes of any triangle ABC intersect at their orthocentre, H.

The orthocentres of the triangles ABH, BCH, ACH are respectively C, A and B.

The four triangles ABC, ABH, BCH and ACH have the same nine-point circle. The four incircles and twelve excircles all touch this nine-point circle.

All four triangles have the same circumradius.

The reflection of H in any side of ABC lies on the circumcircle of ABC, and the same applies to each of the points A, B and C.

The centroids of the four triangles form a set of four points which is similar to the original set ABCH, but one third the size.