December 5, 2005

Outline of properties of Altitudes, the Orthic Triangle, and the Nine-Point Circle

In a triangle ABC, there is a rich set of relationships among the special lines and points of the triangle. For many of the proofs, see the Notes as a webpage (or pdf) These notes include what was emphasized in class, especially items 2, 3, 5 below.

Prior Assignments: Item 4 was observed in the section on angle bisectors. Item 6 was proved as part of median concurrence, but it was reviewed in Lab 10. The important concurrence of altitudes (item 1) was proved by dilation in Lab 10.

Here are some key properties.

  1. Definition: The Nine-Point Circle (or the Euler Circle) of a triangle ABC is the circumcircle of the midpoint triangle A'B'C' of ABC,
  2. Theorem: The centroid G (the point of concurrence of the medians) is the center of the dilation with ratio -1/2 that maps the circumcircle of ABC to the the Nine-Point Circle of ABC. (This is just the 2/3 relationship of the location of the centroid G on each median.)
  3. Theorem. The altitudes of a any triangle ABC are concurrent at a point H. (This was proved in Lab 10 using the dilation of 2 above.
  4. Definitions: The point H of concurrence of the altitudes is called the orthocenter of ABC. The feet of the altitudes of ABC form a triangle called the orthic triangle.
  5. Theorem: If O and B denote the centers of the circumcircle of ABC and the nine-point circle of ABC, then the four points H, B, G, O are collinear in this order, with HB = (1/2)x, BG = (1/6)x, GO = (1/3)x, where x = HO. (This was proved in Lab 10.)
  6. The orthocenter H is the center of the dilation with ratio 1/2 that maps the circumcircle of ABC to the Nine-Point Circle of ABC.
  7. Theorem: If one starts with a triangle XYZ and constructs the interior and angle bisectors that are concurrent at the incenter H and 3 excenters A, B, C, then XYZ is the orthic triangle of the triangle of excenters ABC and the incenter H is the orthocenter. (This was proved, without the orthic terminology as part of the theory of concurrence of angle bisectors.)
  8. Theorem: The altitudes and sides of ABC form the interior and exterior angle bisectors of the orthic triangle. (This is proved in the Notes above and in class 12/5).
  9. The nine-point center passes through 9 special points: the six midpoints of AB, BC, CA, HA, HB, HC, and also the feet of the altitudes of ABC.
  10. Theorem: A triangle EFG is inscribed in triangle ABC, if the vertices E, F, G are each on a different side of ABC. The orthic triangle of ABC has the smallest perimeter of all the triangles inscribed in ABC. (This is proved in class 12/5)