product of integer lengths 6 line segments is a perfect square
Source: Dutch NMO 2010 p3
September 6, 2019
geometryPerfect SquareparallelogramInteger
Problem Statement
Consider a triangle and a point in its interior. Three lines through are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from to the sides of the triangle. The lengths of these segments are integer numbers and (see figure).
Prove that the product is a perfect square.[asy]
unitsize(1 cm);pair A, B, C, D, E, F, O, X, Y, Z;X = (1,4);
Y = (0,0);
Z = (5,1.5);
O = (1.8,2.2);
A = extension(O, O + Z - X, X, Y);
B = extension(O, O + Y - Z, X, Y);
C = extension(O, O + X - Y, Y, Z);
D = extension(O, O + Z - X, Y, Z);
E = extension(O, O + Y - Z, Z, X);
F = extension(O, O + X - Y, Z, X);draw(X--Y--Z--cycle);
draw(A--D);
draw(B--E);
draw(C--F);dot("", A, NW);
dot("", B, NW);
dot("", C, SE);
dot("", D, SE);
dot("", E, NE);
dot("", F, NE);
dot("", O, S);
dot("", X, N);
dot("", Y, SW);
dot("", Z, dir(0));
label("", (A + O)/2, SW);
label("", (B + O)/2, SE);
label("", (C + O)/2, SE);
label("", (D + O)/2, SW);
label("", (E + O)/2, SE);
label("", (F + O)/2, NW);
[/asy]