KöMaL 2022 A837

Let all the edges of tetrahedron $A_1A_2A_3A_4$ be tangent to sphere $S$. Let $a_i$ denote the length of the tangent from $A_i$ to $S$. Prove that $\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\! 2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg).$