FPC25-030.Kite Construction

Summer is coming up, and you and your friends love to fly kites at the beach. You plan to make $n$ new kites this year, and paint them in some range of colours. For this, you already have a practically unlimited amount of canvas and paint, but the limiting factor is where you can put the painted kites while they dry.

To hang out the kites for drying, you found a square field with side length $\ell$. On each of the four sides of the field, there are $n$ poles. The corners of the field are at coordinates $(0, 0)$, $(\ell, 0)$, $(0, \ell)$, and $(\ell, \ell)$, and you know the coordinates of the poles in this coordinate system. There are no poles exactly on a corner of the square, and no poles inside the square.

Since you are planning to make as many kites as there are poles on each side of the field, and each kite has four corners, you want to attach each of the corners of each kite to a pole from each of the four sides. Additionally, each pole should be attached to exactly one corner of a kite. Of course, larger kites are more fun, so you want to maximize the total area of the kites.

Although you vaguely remember from geometry class that a kite has some special properties, you decide to use your artistic freedom and just allow any (convex) quadrilateral as the shape of your kites.