Kissing circles or Soddy circles
Given any three points it is possible to draw circles with centres at these points that just touch each other. It is also possible to draw two other circles that touch these three (one inside all three and one outside all of them.
What are the radii of the circles if the points are the vertices of a 3, 4 ,5 triangle? How about if the triangle was isosceles? Can you calculate the radii of the green and red circles if the points are the vertices of an equilateral triangle? What will happen as the point C moves towards the line given by the points A and B? What happens if it crosses this line? What happens if one point circles one of the other points?