circles are so interesting
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Invert Two Circles Into Equal Ones
Seven and the Eighth Circle Theorem - solution A circle through the center of inversion maps onto a straight line. Thus circle c becomes a straight line under inversion with center X. Circles c1 and c2 are mapped into each other's reflections in this line, which are therefore equal.
Clifford's Circle Theorems - Futility Closet
Let four circles (blue) pass through a single point, M. Each pair of these circles intersect at a second point (pink). Each three of the four blue circles will have three pink points among them; these trios of pink points define four new circles (brown), which intersect in a single point, P. If we start with five circles passing through a single point M, then we can apply the procedure above to each subset of four of them. This will produce five points P that all lie on a single circle. If we
The five circles theorem
The five circles theorem | Curiosa Mathematica Draw five circles, centered on a common sixth circle and intersecting each other chainwise on the same circle (as shown in blue). Now join the second intersection points (as shown in red), in order to form a generally irregular pentagram. Then the vertices of this pentagram lie precisely on the five circles.