InversionProperty

Inversion property

Consider a circle (c) with diameter |FG| = 2b and an arbitrary point E on that diameter. Erect orthogonals (e), (f), (g) to that diameter at E, F and G respectively. Draw circle (m) with the property to be simultaneous tangent to (f), (e), (c) and invert this circle with respect to the circle w(E, r), to obtain circle (m*) with center A and radius (z). Then z = r²/(4b) is independent of the location of E on FG, provided r and b remain constant.
Corollary: If (n) is the circle simultaneously tangent to (e), (g) and (c) then its inverse (n*) with respect to w is congruent to (m*). This remark proves Archimedes' Circles porism illustrated in Archimedes.html .

The proof follows from the remarks:
1) (a/2 + b)² = (b-a/2)² + ED² = > ED² = 2ab.
2) ED EB = r² = > EB = r²sqrt(2ab).
3) z/(a/2) = EB/ED = (r²/sqrt(2ab)) / sqrt(2ab) = r²/(2ab).
4) z = r²/(4b) independent of a! Hence, for fixed b, r and E moving on FG, z is constant.
Switch to the pick-move tool (CTRL+2), catch and move E to see that circles m*, n* remain congruent all the time.

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