## Monday, March 8, 2010

### Fox 260

1. Let take the symetry of the red line on x axis, and suppose that the total red line is not a parabola. it is a circle. then it becomes more easier to see.

2. But if you assume that the red curve is a circle, then you'd be distorting the question. Is that not true?

3. no,circle is just an example.suppose that parabola is a special case of circle. than the quadrilateral becames a trapezoid, circle is JUST an example

4. Oh, I see. I think what you meant was that:
if the red curve is symmetric, than max-area-quadrilateral MUST be symmetric as well, which in this case a trapezoid. Could be an isosceles triangle, but not in this case.

That is general statement -intuitively true- but, it would have been an interesting work if one tries to prove this:
"IF A CURVE IS SYMMETRIC AROUND AN AXIS, THEN INSCRIBED MAX-AREA POLIGON IS, TOO, SYMMETRIC AORUND THE SAME AXIS".

5. yeah , you are good man, thanks

6. while shape becomes more symmetric, it reaches max area, because it fills the circle and looks like circle.think about a hexagon, or octagon

7. area of circle is upperbound for max area of quadrilateral. Like circle, area of under parabola is upper bound too, so the shape should become similar to parabola

8. "IF A CURVE IS SYMMETRIC AROUND AN AXIS, THEN INSCRIBED MAX-AREA POLIGON IS, TOO, SYMMETRIC AROUND THE SAME AXIS".
This statement is NOT true! A counterexample can easily be produced.