The triangle $ABC$ is given. Inside its sides $AB$ and $AC$, the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$. Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$, where $[DEF]$ denotes the content of the triangle $DEF$. (David Hruska, Josef Tkadlec)
Problem
Source: 2020 Czech and Slovak Olympiad III A p2
Tags: geometry, geometric inequality, area of a triangle