Kissing Circles

[BMAD]

Inside an orange unit circle we fit two blue circles of radius 1/2. A yellow circle is tangent to the two blue circles and the inner edge of the orange circle. A green circle inside the orange circle is tangent to one of the blue circles, the yellow circle, and the orange circle. If the orange circle is centered at (0,0) what are the coordinates for the centers of the other four circles? The blue ones should be easy.

Presume the circles are placed in the circle as shown in the picture.

[bonanova]

Center coordinates:

Large: (0, 0)

Halves: (-.5, 0); (.5, 0)

Top: (0, 2/3)

Upper left: (-.5, 2/3)

When four circles are mutually tangent, their radii are related by Descartes theorem.

Applying Descartes first to the large and halves, we get 1/3 as the radius of the top circle.

Applying Descartes next to the large, left and top circles, we get 1/6 as the radius of the small circle.

This centers the top circle a distance of 1/3 unit down from the top of the large circle.

The small circle is thus 1/6 unit above the left and 1/2 unit left of the top circle.

The small circle is thus directly above left and directly left of top.

That locates all the centers.

[BMAD]

consider the radius of the 1/2 circles...how far would that put the centers from the starting center