```{-# LANGUAGE DeriveFunctor #-}
module Data.Geometry.Triangle where

import Data.Bifunctor
import Control.Lens
import Data.Ext
import Data.Geometry.Point
import Data.Geometry.Ball
import Data.Geometry.Properties
import Data.Geometry.Transformation

data Triangle p r = Triangle (Point 2 r :+ p)
(Point 2 r :+ p)
(Point 2 r :+ p)
deriving (Show,Eq)

instance Functor (Triangle p) where
fmap f (Triangle p q r) = let f' = first (fmap f) in Triangle (f' p) (f' q) (f' r)

type instance NumType   (Triangle p r) = r
type instance Dimension (Triangle p r) = 2

instance PointFunctor (Triangle p) where
pmap f (Triangle p q r) = Triangle (p&core %~ f) (q&core %~ f) (r&core %~ f)

instance Num r => IsTransformable (Triangle d r) where
transformBy = transformPointFunctor

-- | Compute the area of a triangle
area   :: Fractional r => Triangle p r -> r
area t = doubleArea t / 2

-- | 2*the area of a triangle.
doubleArea                  :: Num r => Triangle p r -> r
doubleArea (Triangle a b c) = abs \$ ax*by - ax*cy
+ bx*cy - bx*ay
+ cx*ay - cx*by
-- Based on determinant of a 3x3 matrix (shoelace formula)
where
Point2 ax ay = a^.core
Point2 bx by = b^.core
Point2 cx cy = c^.core

-- | get the inscribed disk. Returns Nothing if the triangle is degenerate,
-- i.e. if the points are colinear.
inscribedDisk                  :: (Eq r, Fractional r) => Triangle p r -> Maybe (Disk () r)
inscribedDisk (Triangle p q r) = disk (p^.core) (q^.core) (r^.core)
```