Tangent_Cuts

Envelope of tangent-cuts on circles

Here we consider three circles c₁, c₂, c₃ belonging to the same bundle of circles I(c₁, c₂), generated by the first two of them. Tangents drawn from points P of circle c₁ to c₂, c₃, cut again c₁ respectively at two points A, B. The envelope of line AB is another circle c₄, which belongs also to bundle I(c₁,c₂).

The theorem is a special case of a more general one, valid for conics (see Tangent_Cuts_Envelope.html ). I was searching for an elementary geometric proof, but was unable to find one. There are also interesting analytic relations involved, I would like to know. For example, the determination of the relation f(r₁,r₂,r₃,r₄)=0; satisfied by the radii of the four circles of the bundle.

Remarks
[1] There is a second circle c₅ of the bundle tangent to line BC. But this circle changes in size and location as P moves on c₁. c₄ is distinguished from c₅ by the fact that lines AF, CD, BE intersect at a common point K.
[2] The contact points F, G of the two circles c₄, c₅ with BC are harmonic conjugate with respect to B,C. They are also conjugate with respect to the quadratic transformation defined by the bundle of circles.
[3] There is a position for A, for which lines BC and DE become parallel. Then G falls at infinity and c₅ becomes the radical axis of the bundle. Thus, BC, DE are then parallel to the radical axis of the bundle.

Envelope of tangent-cuts on circles

See Also