In an acute angled triangle \( \mathrm{ABC} \), let \( \mathrm{AD}, \mathrm{BE} \) and \( \mathrm{CF} \) be the perpendiculars from \( \mathrm{A}, \mathrm{B} \) and \( \mathrm{C} \) upon the opposite sides of the triangle. (All symbols used have usual meaning in a triangle.)
\( \mathrm{P} \)
W
The ratio of the product of the side lengths of the triangles \( \mathrm{DEF} \) and \( \mathrm{ABC} \), is equal to
(A) \( \frac{3(a b c)^{\frac{1}{3}}}{4(a+b+c)} \)
(B) \( \frac{1}{4} \)
(C) \( \cos A \cos B \cos C \)
(D) \( \sin \left(\frac{\mathrm{A}}{2}\right) \sin \left(\frac{\mathrm{B}}{2}\right) \sin \left(\frac{\mathrm{C}}{2}\right) \)
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