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What is the maximum area of the inscribed rectangle of a triangle with the base A and the height H? Thank you.
Let △ABC, BC=a, AH=h, and the width of the rectangular EFGH is X and the length is Y.

Similar triangles.

(h-x)/h=y/a

Y = a (h-x)/h is obtained.

The area of rectangular EFGH is

S=xy=x[a(h-x)]/h=(-a/h)x? +ax,∫-a/h & lt; ∴S has a maximum value.

When x=-a/[2(-a/h)]=h/2,

The maximum area of an inscribed rectangle is S=-a? /[4(-a/h)]=ah/4