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an equilateral triangle in a right triangle

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November 15, 2009
geometryrotationPythagorean Theorem

Problem Statement

The figure below shows a right triangle △ABC \triangle ABC. [asy]unitsize(15); pair A = (0, 4); pair B = (0, 0); pair C = (4, 0); draw(A -- B -- C -- cycle); pair D = (2, 0); real p = 7 - 3sqrt(3); real q = 4sqrt(3) - 6; pair E = p + (4 - p)*I; pair F = q*I; draw(D -- E -- F -- cycle); label("AA", A, N); label("BB", B, S); label("CC", C, S); label("DD", D, S); label("EE", E, NE); label("FF", F, W);[/asy] The legs AB‾ \overline{AB} and BC‾ \overline{BC} each have length 4 4. An equilateral triangle △DEF \triangle DEF is inscribed in △ABC \triangle ABC as shown. Point D D is the midpoint of BC‾ \overline{BC}. What is the area of △DEF \triangle DEF?
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