Let
![A_1A_2A_3A_4](/media/m/9/f/c/9fc60bc7746a37e2c1a8fb688ba3a2ea.png)
be a non-cyclic quadrilateral. Let
![O_1](/media/m/7/2/b/72b270d556043f6f393afbf50620eb57.png)
and
![r_1](/media/m/9/0/1/901ecb943995b3585cd44466e1b750cb.png)
be the circumcentre and the circumradius of the triangle
![A_2A_3A_4](/media/m/9/f/1/9f12c32d61e77daaea06ac13b8825a7d.png)
. Define
![O_2,O_3,O_4](/media/m/e/e/7/ee7e3402137ee78dff1aaf8f766881e5.png)
and
![r_2,r_3,r_4](/media/m/8/6/4/864ab7e38475098a853f9a416bc145c7.png)
in a similar way. Prove that
![\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.](/media/m/4/6/1/4614754202d15cb9b35bee8bc55b70f6.png)
Proposed by Alexey Gladkich, Israel
%V0
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
$$\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.$$
Proposed by Alexey Gladkich, Israel