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 $$\overline{BF} ?$$

×