Mathematics IV: circles

[muon2]

The problem is a specific case of Descartes' theorem that says that four mutually tangent circles have curvatures (k_i = 1/r_i where r is the radius) given by the following quadratic equation:

(k₁ + k₂ + k₃ + k₄)² = 2(k₁² + k₂² + k₃² + k₄²)

Since this is a quadratic equation, if three circles are specified then there are at most two solutions for the fourth circle. In this problem one circle is replaced by a straight line which can be considered to be a circle of curvature 0. The two solutions are the small radius inside the "triangle" and the large radius drawn by butafly.

However, the problem said touch without intersecting, but not touch at a single point. If intersecting is taken to mean that one curve has points on either side of another curve (the usual definition), then an object does not intersect itself. In that case the two given circles are also solutions which brings the total to 4.