Inscribed Rectangle
Consider the four vertices of a rectangle. To have a rectangle inscribed in a triangle, two of the rectangle's vertices must lie on the same side of the triangle. Here are some examples:
Given a triangle, construct the inscribed rectangle with maximum area. Is there a "maximum rectangle" for each side of the triangle?
What is the relation of the area of the maximum rectangle to the area of the original triangle? Prove it!
Hint  a drawing and some notation.
Further hint (only if desperately needed): Click Here

Extensions 
Extension Problem: Try this.
Presented by Alan Russell (Guest lecturer).
Submit your idea for an investigation to InterMath.
