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AMC 12 only

Inversion & Spiral Similarity

Two advanced tools for the hardest late-contest geometry problems — rarely essential, but occasionally the only clean way in.

These two techniques show up rarely on the AMC — usually only in the last two or three problems of the AMC 12 — but when a configuration is built from tangent circles or rotation-and-scaling, they can turn an intractable mess into a short computation.

Inversion

An inversion centered at point OO with radius rr sends each point PP(other than OO) to a point P′P' on ray OPOP satisfying:

OP⋅OP′=r2OP \cdot OP' = r^2

Inversion has a few properties that make it powerful: circles and lines map to circles or lines (a circle through the center of inversion maps to a line), and angles between curves are preserved. It's most useful for problems with several mutually tangent circles, since inversion centered at a point of tangency often turns the tangent circles into parallel lines — a much easier configuration to reason about.

Spiral similarity

A spiral similarity is a rotation combined with a scaling (homothety), both centered at the same point. It's the transformation that takes one segment to another non-parallel segment of different length — and a classical result guarantees that whenever two segments ABAB and CDCDmeet (or their extensions do) at a point XX, there is a unique spiral similarity centered at some point mapping A→CA\to C and B→DB \to D, and that center lies on the circumcircles of both △XAD\triangle XAD and △XBC\triangle XBC.

Worked example

Problem: Two circles are internally tangent at point TT. A line throughTT intersects the smaller circle again at AA and the larger circle again atBB. Explain, using inversion, why the tangent lines to the two circles at AAand at BB are parallel.

Solution sketch: Apply an inversion centered at TT. Since both circles pass through the center of inversion, each maps to a line. Because the two original circles were tangent at TT, their image lines are parallel (tangency at the inversion center becomes “never meeting again,” i.e. parallel lines). The tangent-line directions at AAand BB in the original picture correspond to these now-parallel images, which is exactly why the original tangent lines turn out parallel — a fact that's genuinely awkward to prove with angle chasing alone, but nearly immediate with inversion.

Practice problems

1.An inversion has radius r=6r=6 centered at OO. If OP=4OP = 4, find OP′OP'.

2.Under inversion, what happens to a circle that passes through the center of inversion?

3.A spiral similarity maps segment ABAB (length 4) to segment CDCD (length 10), with a rotation of 40°40°. What is the scale factor of the spiral similarity?

4.Why are inversion and spiral similarity considered “AMC 12 only, very light” tools rather than core material?

Geometric Probability