Each of the given $9$ lines cuts a given square into two quadrilateral,whose areas are in ratio $2:3$. Prove that at least three of these lines pass through the same point.
how to approach this question I tried by making possible combinations but didn't seem to get anything helpful. Can anyone prove this rigourously?
Put the square's vertices at $(\pm1,\pm1)$. Since each line splits the square into two quadrilaterals, they individually cut the square at opposite sides. It is then rather easy to show that such a line must pass through one of the four points $(0,\pm4/5),(\pm4/5,0)$ (otherwise they do not partition the area in a $2:3$ ratio). By the pigeonhole principle ($9$ pigeons, $4$ holes), one of those four points must be incident to three different lines.