Question: what is the maximum area of a rectangle inscribed in a triangle?
Answer: If ABC is the triangle, let FG be that line, and FGHK be the rectangle. FG is half the length of AB, and FH, a side of the rectangle, is half the altitude CD of the triangle. So the area of the rectangle is 1/4 the base of the triangle times its height, so the rectangle is half the area of the triangle.