High school students taking geometry are until this day tasked to locate the incircle of a triangle: The circle that touches all sides. One learns that its center is where the three angle bisectors meet, and that’s that.
It’s less often taught that there are three more circles (the excircles), touching two sides of the triangle from the inside, but one form the outside. Their three centers are the corners of a triangle in which the former angle bisectors become the altitudes.
Of course things get really interesting when we move into space. Here the four planes of a tetrahedron can be touched by as many as eight spheres. In the simplest case, it looks like the picture above.
Curiously, this does not work with the regular tetrahedron, it needs to be either more or less elongated.
The Inner Frame 