What is the area of the triangle inside the hexagon?

Let ABCDEF be a convex hexagon, such that FA = AB, BC = CD, DE = EF, and angle FAB = 2 angle EAC. Suppose that the area of triangle ABC is 25, the area of triangle CDE is 10, the area of triangle EFA is 25, and the area of triangle ACE is x. Find, with proof, all possible values of x.

Rotate the hexagon through point A so that B is rotated onto point B’ = F and C is rotated onto C’. Then AC’ = AC, ΔAFC’ = ΔABC, and <C’AE = <EAC (since each are one half of <FAB = <C’AC), hence ΔAC’E = ΔACE Thus, C’E = EC, so ΔE’FE=ΔECD.

Therefore, Δx is composed of ABC + AEF + ECD = 60

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One Response to “What is the area of the triangle inside the hexagon?”

Quadrillerator on September 28th, 2009 at 1:33 am

Rotate the hexagon through point A so that B is rotated onto point B’ = F and C is rotated onto C’. Then AC’ = AC, ΔAFC’ = ΔABC, and <C’AE = <EAC (since each are one half of <FAB = <C’AC), hence ΔAC’E = ΔACE Thus, C’E = EC, so ΔE’FE=ΔECD.

Therefore, Δx is composed of ABC + AEF + ECD = 60
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