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d(A,BC)/BC is rational if AB=AC and A,B,C are lattice points

Source: Netherlands - Dutch NMO 1975, year 1975-76 (also named as 1975 -2) p4

January 27, 2023
geometryanalytic geometrylattice pointsEquilateral

Problem Statement

Given is a rectangular plane coordinate system.
(a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates.
(b) In the plane the vertices A,BA, B and CC lie with integer coordinates in such a way that AB=ACAB = AC. Prove that d(A,BC)BC\frac{d(A,BC)}{BC} is rational.
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