checking if something is a square, in haskell

square.hs

import Data.List

-- checks if a list of points forms a square, by checking:
--   * if |b - a| == |c - a|,
--   * if 0 == (b - a) . (c - a), and
--   * if a + (b - a) + (c - a) == d
--   (a, b, c, d are sorted by distance from a)
isSquare (a:as) = and [
  mag2 (b - a) == mag2 (c - a),
  0 == (b - a) `dot` (c - a),
  b + (c - a) == d
  ] where sortfunc x y = compare (mag2 $ x - a) (mag2 $ y - a)
          [b, c, d] = sortBy sortfunc as

-- a convenience class for dealing with 2d vectors
data Vec a = Vec a a deriving (Eq, Show, Read)

-- addition and subtraction
instance (Num a) => Num (Vec a) where
  (Vec a b) + (Vec c d) = Vec (a + c) (b + d)
  negate (Vec a b) = Vec (-a) (-b)

-- magnitude squared
mag2 (Vec a b) = a*a + b*b

-- dot products
(Vec a b) `dot` (Vec c d) = a*c + b*d

-- main function to read points and check them
main = do
  putStrLn "list of points to check in form [(a, b), ...]: "
  pts <- getLine
  print $ isSquare [Vec a b | (a, b) <- read pts]