Consider the points \(A(0,12), B(10,9), C(8,0),\) and \(D(-4,7).\) There is a unique square \(S\) such that each of the four points is on a different side of \(S.\) Let \(K\) be the area of \(S.\) Find the remainder when \(10K\) is divided by 1000.
(第二十三届AIME1 2005 第14题)