Given a triangle, we wish to inscribe within it a rectangle of largest possible area. How will the area of this rectangle compare with the area of the triangle?
Specifically, let ABC be an arbitrary triangle, with the angles at B and C being acute. Let PQRS be a rectangle with P on side AB, Q and R on side BC, and S on side AC. We wish to find the maximum possible value of area (PQRS) / area (ABC).
This applet allows you to change the shape of the inscribed rectangle by dragging the slider. Use it to see whether the answer is what you intuitively expect.
Note: You can also change the shape of the triangle itself. However it is important that the angles at B and C remain acute.