A simple applet to show that the three angle bisctors of a triangle meet in a single point.

The three vertices of the triangle above can each be dragged to show that the angle bisectors still meet in a single point when we look at other triangles
When a mathematician sees a behavior like this that works with all triangles (or at least with a bunch of examples that we have looked at) the suspicion is that there must be a structure that helps us prove that it must always happen. The slider ProofSteps lets us go through the proof. From any point on an angle bisector, we can drop a perpendicular to the two sides and construct a circle that is tangent to those two sides. Moving down the bisector we reach a point where the circle is tangent to all three sides. That point must be on all three angle bisectors.