Problems

Each number denotes the area of the rectangle it is written into. What number should be written where the "?" character is?

A square was cut with two parallel lines that are $6$ cm apart. One of them goes through the top right corner and the other through the bottom left corner. The three regions obtained this way, two triangles and a parallelogram, have equal areas. What is the area of the square?

The area of the triangle $\mathrm{△}AEC$ is $4$, the area of the triangle $\mathrm{△}BCE$ is $9$, the area of the triangle $\mathrm{△}ABC$ is $21$. What is the area of the triangle $\mathrm{△}ADE$?

A series of squares are connected by touching vertices. Some triangles were drawn outside and inside of the chain. Show that the total green area is the same as the total red area.

In a triangle $\mathrm{△}ABC$, $D$ is the midpoint of $BC$, and $E$ is the midpoint of $AD$. $F$ is the intersection of the side $AC$ with $BE$. What is the area of the triangle $\mathrm{△}AEF$ as a proportion of the area of the triangle $\mathrm{△}ABC$?

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: