Problems

Let $ABCD$ be a parallelogram. The segment $EF$ is parallel to the diagonal $BD$, and the segment $EG$ is parallel to the diagonal $AC$. Show that the areas of the triangles $\mathrm{△}EFD$ and $\mathrm{△}EGC$ are equal.

In a trapezium $ABCD$, the side $AB$ is parallel to the side $CD$. Prove that the areas of triangles $\mathrm{△}ABC$ and $\mathrm{△}ABD$ are equal.

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is $18$, find the area of the small blue triangle.

On the left there is a circle inscribed in a square with side $1$. On the right there are $1$6 smaller, identical circles, which all together fit inside a square of side $1$. Which area is greater, the yellow or the blue one?

In a parallelogram $ABCD$, point $E$ belongs to the side $CD$ and point $F$ belongs to the side $BC$. Show that the total red area is the same as the total blue area:

The figure below is a regular pentagram. What is larger, the black area or the blue area?

A circle is inscribed in a square, and another square is inscribed in the circle. Which area is larger, the blue or the orange one?

In a square, the midpoints of its sides were marked and connected to the vertices of the square. There is another square formed in the centre. Find its area, if the side of the square has length $10$.

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 $AD$. We know that 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 regular hexagon, some diagonals were drawn. Find the area of the red region, if the total area of the hexagon is $72$.