KöMaL 2013 A586

Two convex quadrilaterals have side lengths $a_1$, $a_2$, $a_3$, $a_4$, and $A_1$, $A_2$, $A_3$, $A_4$; their areas are $t$ and $T$, respectively. Show that $16tT \le \sum_{i=1}^4 a_i^2 \big(A_{i+1}^2+A_{i+2}^2+A_{i+3}^2-A_i^2\big) + 4a_1A_2a_3A_4 + 4A_1a_2A_3a_4.$ (The indices are used modulo 4.)