The purpose of this chapter is to recall some of these half-forgotten things to which Dr. Bell referred, to derive some new theorems, developed since Euclid, and to apply our findings to interesting situations. We consider an arbitrary triangle and its most famous associated points and lines: the circumcenter, medians, centroid, angle-bisectors, incenter, excenters, altitudes, orthocenter, Euler line, and nine-point center.

The angle-bisectors lead naturally to a digression on the Steiner–Lehmus theorem, which was believed for a hundred years to be difficult to prove, though we see now that it is really quite easy.

Finally, from a triangle and a point P of general position, we derive a new triangle whose vertices are the feet of the perpendiculars from P to the sides of the given triangle. This idea leads to some amusing developments, some of which are postponed till the next chapter.

The extended Law of Sines

The Law of Sines is one trigonometric theorem that will be used frequently. Unfortunately, it usually appears in texts in a truncated form that is not so useful as an extended theorem could be.