Basic Triangle Centers
From Paul Yiu's MathFest presentation in 2008
Basic Triangle Centers, Circles, and Lines.
Circumcircle, nine-point circle, the Euler line, the circumcenter, centroid, nine-point center, and the orthocenter.
HN : NG : GO = 3 : 1 : 2
The Orthocentroidal Circle
N and O are inverses with respect to the orthocentroidal circle.
Euler Line Reflections
Reflect the Euler line in each side of the triangle. The reflections concur at a point on the circumcircle.
The Fermat Point
Construct equilateral triangles on the exterior of the triangle. The lines AA', BB', and CC' are concurrent. This point is the Fermat point.
The Negative Fermat Point
Construct equilateral triangles on the interior of the triangle. The lines AA', BB', and CC' are concurrent. This point is the Negative Fermat point.