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fixed area of triangle from Fibonacci lattice points

Source: 2017 XX All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p5

May 24, 2022
geometrylattice pointsarea of a triangleareasUkrainian TYM

Problem Statement

The Fibonacci sequence is given by equalities F1=F2=1,Fk+2=Fk+Fk+1,k∈NF_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N. a) Prove that for every m≥0m \ge 0, the area of ​​the triangle A1A2A3A_1A_2A_3 with vertices A1(Fm+1,Fm+2)A_1(F_{m+1},F_{m+2}), A2(Fm+3,Fm+4)A_2 (F_{m+3},F_{m+4}), A3(Fm+5,Fm+6)A_3 (F_{m+5},F_{m+6}) is equal to 0.50.5. b) Prove that for every m≥0m \ge 0 the quadrangle A1A2A4A_1A_2A_4 with vertices A1(Fm+1,Fm+2)A_1(F_{m+1},F_{m+2}), A2(Fm+3,Fm+4)A_2 (F_{m+3},F_{m+4}), A3(Fm+5,Fm+6)A_3 (F_{m+5},F_{m+6}), A4(Fm+7,Fm+8)A_4 (F_{m+7},F_{m+8}) is a trapezoid, whose area is equal to 2.52.5. c) Prove that the area of ​​the polygon A1A2...AnA_1A_2...A_n , n≥3n \ge3 with vertices does not depend on the choice of numbers m≥0m \ge 0, and find this area.
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