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)