Circles around

Given a triangle ABC, we construct three congruent circles (Da), (Ea), (Fa) such that

  • (Da) is tangent to AB, BC and (Ea),
  • (Ea) is tangent to BC, (Da) and (Fa)
  • (Fa) is tangent to BC, CA and (Ea)

We construct circles (Db), (Eb), (Fb) and (Dc), (Ec), (Fc) in a similar way. Then we have:

1. Lines AEa, BEb and CEc are concurrent.

2. Points Da, Fa, Db, Fb, Dc, Fc lie on the same conic, an ellipse.
The same is true for any number of circles:

In any case the perspector is

and the radii of the circles tangent to the side BC are equal to

where S stands for twice the area of triangle ABC.

Remark: These formulas are valid when n=1, when each circle tangent to a side is the corresponding excircle of the triangle.

 

References: