VMO 1969 Problem 3

Consider $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each $i = 1, ... ,4$ we have $(x_i -a)^2 +(y_i -b)^2 \le c^2$ . Prove that $a^2 + b^2 < c^2$ . Restate this fact in the form of geometric result in plane geometry.