MidCircle2

Mid circle of two tangent circles

Consider two tangent circles {a(A, ra), b(B, rb)}. Draw two parallel radii AE, BF from their centers. The intersection point C of lines {EF, AB} determine one [similarity center of the two circles]. The ratio being AC/BC = ra/rb.
-Extend AG and BF defining H. Then triangle HGF is isosceles and circle h(H) is tangent externally to both a and b.
-Define circle c(C) by the property to be centered at C and orthogonal to h(H). The inversion w.r. to c interchanges {a,b}.
-The radius rc of c(C) is independent of the particular h(H). c(C) is orthogonal to every circle tangent to both {a,b} externally or internally (both). c belongs to the coaxal bundle of circles tangent at the common point of a and b.
-c(C) is called the [Mid-circle] of a and b. See the file MidCircles.html for other configurations of the two circles.

When the circles are disjoint there is also one mid circle inverting one to the other. When they are intersecting there are two mid circles, see the file MidCircles.html discussing this case.

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