Nine Point Circle
Drag the corners of the triangle to move it around.
Given a triangle, consider the following (seemingly unrelated) sets of points:
  • The three midpoints of the sides (light blue dots, to the right)
  • The feet of the altitudes (red dots to the right)
  • The midpoints of the lines from each triangle-vertex to the orthocenter (yellow dots to the right)
Here's the magic - these three sets of points lie on a single circle! Play with the demonstration to get an intuition for this, or if you're feeling up for it, check out wikipedia for a proof.