Rectangle in a Triangle

Geometry Level 3

Triangle \( ABC \) has side lengths \( \overline{AB} = 10\), \(\overline{BC} = 20 \), and \( \angle B = 90^{\circ} \) with three points \( D,\ E \) and \( F \) lying on sides \( BC,\ CA \) and \( AB \), respectively. If \( BDEF \) is a rectangle with area less than or equal to \( 32 \) and \( \overline{BF} > 5 ,\) what is the minimum value of \( \overline{BF} ?\)

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