hexagons with the same area
Source: 1st National Women's Contest of Mexican Mathematics Olympiad 2022 , level 1 p1
July 22, 2023
Mexicogeometryalgebra
Problem Statement
There are sticks of each color between blue, red and green, such that we can make a triangle with sides sticks with all different colors. Dana makes two arrangements, she starts with and uses the other six sticks to extend the sides of , as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same.[asy]size(300);
pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K;
A = (0, 0);
B = (1, 0);
C=(-0.5,2);
D=(-1.1063,4.4254);
M=(-1.7369,3.6492);
N=(3.5,0);
P=(-2.0616,0);
Q=(0.2425,-0.9701);
R=(1.6,-0.8);
S=(7.5164,0.8552);
T=(8.5064,0.8552);
U=(7.0214,2.8352);
V=(8.1167,-1.546);
W=(9.731,-0.7776);
X=(10.5474,0.8552);
Y=(6.7813,3.7956);
Z=(6.4274,3.6272);
K=(5.0414,0.8552);
draw(A--B, blue);
label("", (A + B) / 2, dir(270), fontsize(10));
label("", (B+C) / 2, dir(10), fontsize(10));
label("", (A+C) / 2, dir(230), fontsize(10));
draw(B--C,green);
draw(D--C,green);
label("", (C + D) / 2, dir(10), fontsize(10));
draw(C--A,red);
label("", (C + M) / 2, dir(200), fontsize(10));
draw(B--N,green);
label("", (B + N) / 2, dir(70), fontsize(10));
draw(A--P,red);
label("", (A+P) / 2, dir(70), fontsize(10));
draw(A--Q,blue);
label("", (A+Q) / 2, dir(540), fontsize(10));
draw(B--R,blue);
draw(C--M,red);
label("", (B+R) / 2, dir(600), fontsize(10));
draw(Q--R--N--D--M--P--Q, dashed);
draw(Y--Z--K--V--W--X--Y, dashed);
draw(S--T,blue);
draw(U--T,green);
draw(U--S,red);
draw(T--W,red);
draw(T--X,red);
draw(S--K,green);
draw(S--V,green);
draw(Y--U,blue);
draw(U--Z,blue);
label("", (Y+U) / 2, dir(0), fontsize(10));
label("", (U+Z) / 2, dir(200), fontsize(10));
label("", (S+T) / 2, dir(100), fontsize(10));
label("", (S+U) / 2, dir(200), fontsize(10));
label("", (T+W) / 2, dir(70), fontsize(10));
label("", (T+X) / 2, dir(70), fontsize(10));
label("", (U+T) / 2, dir(70), fontsize(10));
label("", (S+K) / 2, dir(70), fontsize(10));
label("", (V+S) / 2, dir(30), fontsize(10));[/asy]