with the cirlce b passing through A and its diametral point define on circle b a diameter DE orthogonal to line a.

That DE is a diameter is clear, since angle ACB = ECD is a right one. That DE is also orthogonal to a follows by the

angle equalities DEC = DCF, CAB = FCB, DCF+FCB = π/2.

called

the two given circles. Every circle of the transverse system separates the two given circles.

The figure below displays two circles {c, c'} from the direct and transverse system correspondingly. Circles {c, c'} have a common

contact point with k

[1] Line AB which is the line of contacts of c (direct) passes through the outer center S

[2] Line AE which is the line of contacts of c'(transverse) passes through the inner center S

[3] The radical axes of all pairs of circles from {k

[4] Lines {AB,AE} intersect circle k

[5] The centers of the circles of the direct system generate a hyperbola with foci at the centers {O

and major axis |r

[6] The centers of the circles of the transverse system generate an ellipse with foci at the centers {O

circles and major axis equal to |r

[1, 2] is proved in section-6 of RadicalAxis.html . [3] is proved in section-4 of the same file. [4] is proved here in section-1 and

[5, 6] are trivial verifications of the definitions of central conic sections through their basic focal properties.

{k

In this case both the direct and transverse system of tangent circles are ellipses. The direct system consists of ellipses c with foci

at {O

|r

{k

circle k

to line k

from k

circle k

to line k

the two systems are indistinguishable.

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