OPM 2012 Problem 5

Let $[ABC]$ be a triangle. Points $D$ , $E$ , $F$ and $G$ are such $E$ and $F$ are on the lines $AC$ and $BC$ , respectively, and $[ACFG]$ and $[BCED]$ are rhombus. Lines $AC$ and $BG$ meet at $H$ ; lines $BC$ and $AD$ meet at $I$ ; lines $AI$ and $BH$ meet at $J$ . Prove that $[JICH]$ and $[ABJ]$ have equal area.