Find a shape for which the area of the largest inscribed triangle is equal to the area of the largest inscribed square.

No doubt there are an infinity of such shapes, but you must find one that can be expressed by a simple equation of order 2 or less, combined with constraints on the domain and range.

I had a nice sketch of this problem.

I considered that an algebraic solution might be possible but I haven't gotten there.

Knowing this was not the solution, but feeling that it might offer me an approach, I considered a circle of unit radius on an x,y Cartesian grid.

The coords of the vertices of the inscribed square were abs(1/√2) for both x and y.  The area of the square was 2 square units.

The inscribed triangle had its apex at (0,1) and its base subtended from the vertices of the opposing side of the square.  Its area was
 1 + 1/√2   ---- 1.71....

Now, let the vertices of the square remain fixed.  As "+y" decreases let "+x" increase so that the circle is distorted and so becoming an ellipse.

Now, somewhere between y=1 and 1/√2 we have a value for the coordinate set of the triangle which will, along with the coords of the square determine the curve (ellipse) that is required.

I can go no further.

Posted by brianjn on 2007-06-12 11:08:52