[alogo] 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).

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

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.

[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.

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