Similarity_Centers

Similarity centers

Consider two circles a(A, r), b(B, r'). Their common tangents intersecting at two points C and D. The following facts are true.
(1) C, D are similarity centers, the similarity ratio being k = r/r'.
(2) C, D are harmonic conjugate to A, B.
(3) On every line CG intersecting both circles, AH || BG, AJ || BI, and EFIH is cyclic.
(4) The tangent CL to the circumcircle (c) of EFIH has constant length |CL|² = |CE||CF| = R².
(5) The circle f(C, R) defines an [inversion] interchanging circles a and b. All circles (c) are orthogonal to (f).
(6) A circle (e) tangent to both circles (a), (b), as shown, has the chord HI of tangent points passing through C.
(7) Circle (e) is also orthogonal to (f).
(8) The common tangents of b, e at I and a, e at H intersect on the radical axis g of circles a, b.
(9) The lines HE and IF, as well as NI, MH intersect on the radical axis g of circles a, b.
(10) The inverse of P with respect to (f) is D.
(11) The circle with diameter CD, which is the inverse on (f) of the radical axis g, is the locus (Apollonius) of points the distances of which from A and B are at ratio k = r/r'.

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