Exploring the Euclidean Geometry of the Triangle

Roy Wright
This is a visual glossary of various objects associated with any triangle in Euclidean geometry. (The notation used here is meant to match Section 5.6 of Venema's Foundations of Geometry.)
The vertices A, B, and C may be moved around at will. There are many beautiful facts to be discovered, at all levels of difficulty (from fairly simple to very deep), about the objects illustrated here. For example, here are just a few things to notice:
  • The centroid has a famous physical significance -- it's sometimes known as the "center of balance." How can we be sure of this property?
  • Why can we be sure that the three perpendicular bisectors will always meet in a single point?
  • Display the centroid, orthocenter, and circumcenter at the same time. What do you notice about them? And what about the incenter?