MathDB

8

Part of 2023 Mexican Girls' Contest

Problems(1)

hexagons with the same area

Source: 1st National Women's Contest of Mexican Mathematics Olympiad 2022 , level 1 p1

7/22/2023
There are 33 sticks of each color between blue, red and green, such that we can make a triangle TT with sides sticks with all different colors. Dana makes 22 two arrangements, she starts with TT and uses the other six sticks to extend the sides of TT, 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("bb", (A + B) / 2, dir(270), fontsize(10)); label("gg", (B+C) / 2, dir(10), fontsize(10)); label("rr", (A+C) / 2, dir(230), fontsize(10)); draw(B--C,green); draw(D--C,green); label("gg", (C + D) / 2, dir(10), fontsize(10)); draw(C--A,red); label("rr", (C + M) / 2, dir(200), fontsize(10)); draw(B--N,green); label("gg", (B + N) / 2, dir(70), fontsize(10)); draw(A--P,red); label("rr", (A+P) / 2, dir(70), fontsize(10)); draw(A--Q,blue); label("bb", (A+Q) / 2, dir(540), fontsize(10)); draw(B--R,blue); draw(C--M,red); label("bb", (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("bb", (Y+U) / 2, dir(0), fontsize(10)); label("bb", (U+Z) / 2, dir(200), fontsize(10)); label("bb", (S+T) / 2, dir(100), fontsize(10)); label("rr", (S+U) / 2, dir(200), fontsize(10)); label("rr", (T+W) / 2, dir(70), fontsize(10)); label("rr", (T+X) / 2, dir(70), fontsize(10)); label("gg", (U+T) / 2, dir(70), fontsize(10)); label("gg", (S+K) / 2, dir(70), fontsize(10)); label("gg", (V+S) / 2, dir(30), fontsize(10));
[/asy]
Mexicogeometryalgebra