Question 49
The quadrilateral $A_1A_2A_3A_4$ is not cyclic. Let $O_1,r_1$ be the circumcenter and circumradius of triangle $A_2A_3A_4$, and define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ similarly. Prove that
$$\frac1{O_1A_1^2-r_1^2}+\frac1{O_2A_2^2-r_2^2}+\frac1{O_3A_3^2-r_3^2}+\frac1{O_4A_4^2-r_4^2}=0.$$
Illustration only; the configuration/coloring may vary.
$$\frac1{O_1A_1^2-r_1^2}+\frac1{O_2A_2^2-r_2^2}+\frac1{O_3A_3^2-r_3^2}+\frac1{O_4A_4^2-r_4^2}=0.$$
Illustration only; the configuration/coloring may vary.