Given real numbers
![x_1,x_2,y_1,y_2,z_1,z_2](/media/m/e/9/e/e9eb635b4942e998769789eaabc841ae.png)
satisfying
![x_1>0,x_2>0,x_1y_1>z_1^2](/media/m/9/4/3/9436003efd2037429e519d25629cf01b.png)
, and
![x_2y_2>z_2^2](/media/m/3/7/f/37f253d694122cbfdc7e0ff0628177c4.png)
, prove that:
![{8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}.](/media/m/4/7/9/479bc22a12c44c7c2d0b6c50aadae4b3.png)
Give necessary and sufficient conditions for equality.
%V0
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: $${8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}.$$ Give necessary and sufficient conditions for equality.