Reim’s theorem

I was looking at Reim’s theorem the other day: you can read more at Jean-Louis Ayme’s fantastic website http://jl.ayme.pagesperso-orange.fr/ but only if you know French! Here is the basic statement:

Theorem (Reim). Let \omega_1, \omega_2 be two circles intersecting at M,N. Let line \ell_M through M intersect \omega_1, \omega_2 at A_1, A_2. Let B_1, B_2 be points on \omega_1, \omega_2 respectively, Then A_1B_1\parallel A_2B_2 if , and only if, B_1,N,B_2 are collinear on a line \ell_N.

Proof. Suppose that B_1NB_2 is a straight line. Then \measuredangle MA_1B_1 = \measuredangle MNB_1 = \measuredangle MA_2B_2 \implies A_1B_1 \parallel A_2B_2.

The reader is left to prove the reverse implication.

The delight of the theorem is in its many converses and special cases, applicable to a wide range of geometric figures. Again, the reader is invited to be inventive enough to think of some (e.g. what happens if A_1 \equiv M?)

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