The mathematics of a close-packed cluster of 3 cones

Consider a pair of identical right circular cones in close contact along the full length of their curved surfaces and with their vertices at the same point in space. Imagine the axes of these cones to be in the horizontal plane. A third identical cone is then rested on top of the other two such that their curved surfaces are touching at points on circles perpendicular to their axes at around (or exactly) the mid-heights of the cones. The vertex of the third cone is opposite to that of the other two (i.e. this cone is in an inverted attitude with respect to the others).

What I wish to solve is the points of contact of the cones and the straight line equation for the third cone.

I'm not sure of the level of mathematics required to solve this - hopefully elliptic functions are not required!

Any help would be appreciated.

Kind Regards,

Tennessee