Problems

In a parallelogram $ABCD$, point $E$ belongs to the side $AB$, point $F$ belongs to the side $CD$ and point $G$ belongs to the side $AD$. What is more, the marked red segments $AE$ and $CF$ have equal lengths. Prove that the total grey area is equal to the total black area.

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 $\mathrm{△}ABE$ and $\mathrm{△}BCD$ 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 $S_{ABE}+S_{CDE}=S_{BCE}+S_{ADE}$.