Thread: Maximum area of a rectangle inscribed in a triangle

This one is a killer. If someone can prove it, i can get extra credit in my calc class!

"Prove that for any triangle, the inscribed rectangle with a base along one of the triangle's sides will have an area equal to half of the triangle's area."



I know that it's easy to prove visually by folding a piece of paper to look like that above, but algebraically, i don't know how.

Thanks!

Say the triangle is ABC. Let the midpoint of AB be P and the midpoint of AC be Q. Drop perpendiculars from P to BC, meeting at point R, and from Q to BC, meeting at point S. Then the rectangle PQSR has half the area of the triangle.

That works, but it doesn't involve algebra or calculus... unfortunately. We went over how to visually or geometrically prove it in class, but the point is to prove it using calculus and algebra.

I heard that proving this problem with calculus involves using derivatives and areas of the line segments of the figure if it was on a coordinate plane. Does this help?