Consider a triangle XYZ and a point O in its interior. Three lines through O are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from O to the sides of the triangle. The lengths of these segments are integer numbers a,b,c,d,e and f (see figure).
Prove that the product a⋅b⋅c⋅d⋅e⋅f 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", A, NW);
dot("B", B, NW);
dot("C", C, SE);
dot("D", D, SE);
dot("E", E, NE);
dot("F", F, NE);
dot("O", O, S);
dot("X", X, N);
dot("Y", Y, SW);
dot("Z", Z, dir(0));
label("a", (A + O)/2, SW);
label("b", (B + O)/2, SE);
label("c", (C + O)/2, SE);
label("d", (D + O)/2, SW);
label("e", (E + O)/2, SE);
label("f", (F + O)/2, NW);
[/asy] geometryPerfect SquareparallelogramInteger