[ Re-grouping areas $]$
[ Properties of symmetries and axes of symmetry ]

Authors: Galperin G.A., Vasilyev N.B.

$F$ - a convex figure with two mutually perpendicular axes of symmetry. Through a point $M$, lying inside the figure and at distances $a$ and $b$ from the axes, lines parallel to the axes are drawn. These lines divide $F$ into four regions. Find the difference between the sum of the areas of the largest and smallest regions and the sum of the areas of the other two.
Reflecting region $C M D$ (see fig.) with respect to the vertical axis of symmetry, and region $A M B$ - with respect to the horizontal axis, we will overlay them on the larger region $A M D$.

![](https://cdn.mathpix.com/cropped/2024_05_06_cdc67dfeb844af03d94fg-28.jpg?height=437&width=581&top_left_y=1411&top_left_x=750)

In doing so, a rectangle $K L M N$ will remain uncovered, while the segment $E K F$ (equal to the smaller region $B M C$ due to symmetry) will be covered twice. Therefore,

$\left(S_{A M D}+S_{B M C}\right)-\left(S_{A M B}+S_{C M D}\right)=S_{K L M N}=4 a b$.

## Answer

$4 a b$.