MathDB
cartesian triangles

Source: French MO 2000 Problem

April 9, 2021
geometryTriangle

Problem Statement

In this problem we consider so-called cartesian triangles, that is, triangles ABCABC with integer sides BC=a,CA=b,AB=cBC=a,CA=b,AB=c and ∠A=2π3\angle A=\frac{2\pi}3. Unless noted otherwise, △ABC\triangle ABC is assumed to be cartesian.
(a) If U,V,WU,V,W are the projections of the orthocenter HH to BC,CA,ABBC,CA,AB, respectively, specify which of the segments AUAU, BVBV, CWCW, HAHA, HBHB, HCHC, HUHU, HVHV, HWHW, AWAW, AVAV, BUBU, BWBW, CVCV, CUCU have rational length. (b) If II is the incenter, JJ the excenter across AA, and P,QP,Q the intersection points of the two bisectors at AA with the line BCBC, specify those of the segments PBPB, PCPC, QBQB, QCQC, AIAI, AJAJ, APAP, AQAQ having rational length. (c) Assume that bb and cc are prime. Prove that exactly one of the numbers a+b−ca+b-c and a−b+ca-b+c is a multiple of 33. (d) Assume that a+b−c3c=pq\frac{a+b-c}{3c}=\frac pq, where pp and qq are coprime, and denote by dd the gcd⁡\gcd of p(3p+2q)p(3p+2q) and q(2p+q)q(2p+q). Compute a,b,ca,b,c in terms of p,q,dp,q,d. (e) Prove that if qq is not a multiple of 33, then d=1d=1. (f) Deduce a necessary and sufficient condition for a triangle to be cartesian with coprime integer sides, and by geometrical observations derive an analogous characterization of triangles ABCABC with coprime sides BC=aBC=a, CA=bCA=b, AB=cAB=c and ∠A=π3\angle A=\frac\pi3.
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