Day 29

Math 444, Monday, 12/5

1. The altitudes of a any triangle ABC are concurrent at a point H called the orthocenter of ABC.
2. The feet of the altitudes of ABC form a triangle called the orthic triangle.
3. The altitudes and sides of ABC form the interior and exterior angle bisectors of the orthic triangle.
4. 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.
5. The orthocenter H is the center of the dilation with ratio 1/2 that maps the circumcircle of ABC to the circumcircle of the orthic triangle. This circle is called the Nine-Point Circle or the Euler Circle of ABC.
6. 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.
7. 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.
8. 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 all on a line called the Euler Line, in this order, with HB = (1/2)x, BG = (1/6)x, GO = (1/3)x, where x = HO.

Read the web pages linked to the Reading Assignment Below