2005 AIME I Problem 14

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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.