A NOTE ON REYES’S THEOREM ABOUT TRIANGLES

Don Zagier

In the last issue of Nieuw Archief, the following elegant theorem was stated by Reyes [1]:

Theorem. Let A_{1}A_{2}A_{3} be a triangle. We extend the numbering cyclically by A_{i+3} = A_{i} for i ∈ ℤ. Suppose given for each i ∈ ℤ a point X_{i} on the perpendicular bisector of A_{i−1}A_{i} such that the three points X_{i}, A_{i}, X_{i+1} are collinear for each i. Then the sequence {X_{i}} has period 6.

(In the notation of [1] the three vertices A_{1}, A_{2}, A_{3} were denoted by A, B, C and the first four points X_{1}, X_{2}, X_{3} and X_{4} by X, Y, Z and X′, and the result was stated in the form that the map X ↦ X′ from the perpendicular bisector of CA to itself is an involution.) The proof in [1], though described as “quite simple,” is fairly intricate and uses arguments from both vector algebra and trigonometry. Here is a much shorter geometric proof. Let O be the meeting point of the perpendicular bisectors of the sides (= center of the circumscribed circle) of the triangle A_{1}A_{2}A_{3}. The point X_{i} determines, and is determined by, the angle θ_{i} = ∠OA_{i−1}X_{i}. Since X_{i} is on the bisector of the angle A_{i−1}OA_{i}, we have ∠OA_{i}X_{i} = θ_{i}, and since X_{i}, A_{i} and X_{i+1} are collinear we have ∠OA_{i}X_{i+1} = π − ∠OA_{i}X_{i} or θ_{i+1} = π − θ_{i}. Thus the angles θ_{i} alternate between two values θ_{1} and π − θ_{1}, so θ_{i+6} = θ_{i} and X_{i+6} = X_{i}.

[1] W. Reyes, 1996, On a theorem in circle geometry, Nieuw-Arch.-Wisk. 14, no. 2, pp. 231–233.