In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.
The figure below is a regular pentagram. What is larger, the black area or the blue area?
A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?
Let \(E\) and \(F\) be the midpoints of the sides \(BC\) and \(AD\) of the parallelogram \(ABCD\). Find the area of the quadrilateral formed by the lines \(AE, ED, BF\) and \(FC\), if it is known that the area \(ABCD\) is equal to \(S\).
A polygon is drawn around a circle of radius \(r\). Prove that its area is equal to \(pr\), where \(p\) is the semiperimeter of the polygon.
The point \(E\) is located inside the parallelogram \(ABCD\). Prove that \(S_{ABE} + S_{CDE} = S_{BCE} + S_{ADE}\).