Circle Inversion Fractals

Properties of Inversion

The arc determined by angle AED and the arc determined by angle ABD are complementary, so they add to 2*pi.

The measures of AED and ABD are one half the measures of the arcs they determine, consequently

AED + ABD = pi

Also, note AED + DEC = pi.

Consequently, ABD = DEC.

Because triangles ABC and DEC share the angle BCA, they are similar.

Ratios of corresponding sides are equal, so

CD/CA = CE/CB.

and

CD*CB = CA*CE

That is, for any point C outside a circle, and any line through C intersecting the circle in two points, P and Q (B and D, or A and E in the figure), the product CP*CQ is a constant. Call the constant k.

Return to Circles invert to circles.