Find the dimensions of the rectangle of maximum area which can be inscribed in an equilateral triangle as shown in the figure

Question

I do not know what I should start by doing. Could you please help me to start solving this problem, give me a hint or two?

This link talks about something related, but I do not really understand what they are talking about (probably beacause I haven't read enough math).

NOTE: I only want hints or a start of solving this problem.

Thanks a lot.

Answer 1

1. Start by inserting a coordinate system.
2. Now assume a point on the lines where the corners will be. Let's call the point $p$.
3. Now figure out a way to write $p$ as a function of coordinates which follows the triangle sides (a hint could be the equation of a line which you probably have learned about).
4. Express the area as a function of the corner coordinate found in 3.
5. Try and find a way to calculate the maximum of the area function.

Answer 2

There is a theorem that says that if you have a point on the boundary of an equilateral triangle, then the sum of the projections of the point to the other two sides remains constant. Use that theorem and the MA-MG inequality.

Answer 3

Fold the triangle along the edges of the rectangle (three folds). This way, the three small triangle parts will always cover the rectangle (you need to show this, of course). But if you study this covering, there is one rectangle that is just barely covered by the triangles with no overlap between the triangles and no parts of the triangles outside the rectangle.

This means that this rectangle has an area which is half that of the large triangle and that it is clearly optimal (for any other rectangle, the three triangles have more total area than the rectangle, rather than equal).