Let the triangle have side length x.
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.
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.