The Nine-point
circle, from mathworld, is the circle that passes through the "feet" of
the three altitudes of a triangle. It also passes through the midpoints
of the three sides, and the three midpoints of the segments connecting the
vertices to the orthocenter; hence "nine points".
Point and Triangle gives a
clever method of telling whether a point is inside a triangle. An
algorithm to do this is written in several different languages.
The Triangle Inequality
proves that |a+b|<=|a|+|b| and |a-b|>=|a|-|b|
Triangle Circles
-- Inscribed Circle and Angle Bisectors, with bonus features: exscribed
circles and nine-point circle