Center by Color
Scott Farrar has a blogpost about a cool triangle inquiry activity using GeoGebra's dynamic colors, and that directly inspired this.
http://scottfarrar.com/blog/neighborhoodofthespecialcase/
It also reminds me of Dan Meyer's Taco Cart applet (which is closest). No context, but I think these color inquiries could lead to some good geometry about how to find these special points.
These are different ways to think about the middle of a triangle. When might they be useful? How could you find these special points using the geometry of the triangle? (Angles, distances, bisectors, altitudes...)