Let the triangle have side lengthx.

The first circle will touch at a points that defines a radii perpendicular to the triangle edges. We can derive a formula for the radius of the first circle using the following:

where is the distance from the vertex of a triangle to the circle, is the radius of te circle, and is the height of the triangle.

Also

Use these two equations to eliminate the variable and arrive at:

.

Also observe that can be used to define the height of new equilateral triangles.

.

Then the side length of these new equilaterial trainges in which the next circles will be embedded are:

Now we have a new side length of and we can repeat the procedure to determine the radius of the second circles. By our formula above they will be:

And so on.