Let A

_{1}, A

_{2}, A

_{3}be three points in the plane, and for convenience, let A

_{4}= A

_{1}, A

_{5}= A

_{2}. For n = 1, 2, and 3, suppose that B

_{n}is the midpoint of A

_{n}A

_{n+1}, and suppose that C

_{n}is the midpoint of A

_{n}B

_{n}. Suppose that A

_{n}C

_{n+1}and B

_{n}A

_{n+2}meet at D

_{n}, and that A

_{n}B

_{n+1}and C

_{n}A

_{n+2}meet at E

_{n}. Calculate the ratio of the area of triangle D

_{1}D

_{2}D

_{3}to the area of triangle E

_{1}E

_{2}E

_{3}.

[APMO 1989 Q 3]

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