2005 AIME I Problems/Problem 14
Problem
Consider the points and
There is a unique square
such that each of the four points is on a different side of
Let
be the area of
Find the remainder when
is divided by 1000.
Solution
Let denote a normal vector of the side containing
. The lines containing the sides of the square have the form
,
,
and
. The lines form a square, so the distance between
and the line through
equals the distance between
and the line through
, hence
, or
. With
we get
and
. So the side of the square is
, the area is
, and the answer to the problem is
.