Question 19
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcenter and circumradius of triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ similarly. Prove that
$$\begin{aligned}
&\frac1{O_1A_1^2-r_1^2}+\frac1{O_2A_2^2-r_2^2}\\
&\qquad+\frac1{O_3A_3^2-r_3^2}+\frac1{O_4A_4^2-r_4^2}=0.
\end{aligned}$$
Illustrative configuration only.
$$\begin{aligned}
&\frac1{O_1A_1^2-r_1^2}+\frac1{O_2A_2^2-r_2^2}\\
&\qquad+\frac1{O_3A_3^2-r_3^2}+\frac1{O_4A_4^2-r_4^2}=0.
\end{aligned}$$
Illustrative configuration only.