MathDB

1

Part of 1982 Canada National Olympiad

Problems(1)

Show that the area of A'B'C'D' is twice that of ABCD

Source: Canada National Mathematical Olympiad 1982 - Problem 1

9/30/2011
In the diagram, OBiOB_i is parallel and equal in length to AiAi+1A_i A_{i + 1} for i=1i = 1, 2, 3, and 4 (A5=A1A_5 = A_1). Show that the area of B1B2B3B4B_1 B_2 B_3 B_4 is twice that of A1A2A3A4A_1 A_2 A_3 A_4.
[asy] unitsize(1 cm);
pair O; pair[] A, B;
O = (0,0); A[1] = (0.5,-3); A[2] = (2,0); A[3] = (-0.2,0.5); A[4] = (-1,0); B[1] = A[2] - A[1]; B[2] = A[3] - A[2]; B[3] = A[4] - A[3]; B[4] = A[1] - A[4];
draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[2]--B[3]--B[4]--cycle); draw(O--B[1]); draw(O--B[2]); draw(O--B[3]); draw(O--B[4]);
label("A1A_1", A[1], S); label("A2A_2", A[2], E); label("A3A_3", A[3], N); label("A4A_4", A[4], W); label("B1B_1", B[1], NE); label("B2B_2", B[2], W); label("B3B_3", B[3], SW); label("B4B_4", B[4], S); label("OO", O, E); [/asy]
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