#lang racket/base

(define (square x) (* x x))

(define (lengths_eq x1 y1 x2 y2 x3 y3 x4 y4)
(if (equal? (+ (square(- x1 x2)) (square(- y1 y2))) 
(+ (square(- x3 x4)) (square(- y3 y4)))) 
(displayln "Lines (1,2) and (3,4) have the same length.") 
(displayln "Lines (1,2) and (3,4) do not have the same length.")))

(define (collinear x1 y1 x2 y2 x3 y3)
(if (equal? (* (- x1 x2) (- y2 y3)) (* (- y1 y2) (- x2 x3))) 
#t #f
))

(define (perpendicular x1 y1 x2 y2 x3 y3 x4 y4)
(if (equal? (+ (* (- x1 x2) (- x3 x4)) (* (- y1 y2) (- y3 y4))) 0 ) 
#t #f
))

(define (parallel x1 y1 x2 y2 x3 y3 x4 y4)
(if (equal? (* (- x1 x2) (- y3 y4)) (* (- y1 y2) (- x3 x4))) 
(displayln "Lines (1,2) and (3,4) are parallel.") 
(displayln "Lines (1,2) and (3,4) are not parallel.")))


(define (is_intersection x1 y1 x2 y2 x3 y3 x4 y4 x5 y5)
(if (and (equal? (* (- x1 x2) (- y2 y3)) (* (- y1 y2) (- x2 x3)) ) 
(equal? (* (- x1 x4) (- y4 y5)) (* (- y1 y4) (- x4 x5)) ) ) 
(displayln "Lines (2,3) and (4,5) meet at point 1.") 
(displayln "Lines (2,3) and (4,5) do not intersect at point 1.")))

(define (is_midpoint x1 y1 x2 y2 x3 y3)
(if (and (equal? (* 2 x1) (+ x2 x3)) (equal? (* 2 y1) (+ y2 y3)))
(displayln "Point 1 is the midpoint of line (2,3).")
(displayln "Point 1 is not the midpoint of line (2,3).")
))


(define (grobner_decide a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y)
(if (and (parallel a_x a_y b_x b_y d_x d_y c_x c_y) 
    (parallel a_x a_y d_x d_y b_x b_y c_x c_y)
    (is_intersection e_x e_y a_x a_y c_x c_y b_x b_y d_x d_y))
  (begin 
  (lengths_eq a_x a_y e_x e_y e_x e_y c_x c_y) 
  (displayln "This is a parallelogram.") 
  )
  (displayln "This is not a parallelogram.") 
))


(define (grobner_decide_coll a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y)
(if (and (parallel a_x a_y b_x b_y d_x d_y c_x c_y) 
    (parallel a_x a_y d_x d_y b_x b_y c_x c_y)
    (is_intersection e_x e_y a_x a_y c_x c_y b_x b_y d_x d_y) 
    (not (collinear a_x a_y b_x b_y c_x c_y))
    )
  (begin 
  (lengths_eq a_x a_y e_x e_y e_x e_y c_x c_y) 
  (displayln "This is a parallelogram.") 
  )
  (displayln "This is not a parallelogram.") 
))


(define (grobner_decide_mid a_x a_y b_x b_y c_x c_y d_x d_y m_x m_y)
(if (and (is_midpoint m_x m_y a_x a_y c_x c_y) 
    (not (perpendicular a_x a_y c_x c_y m_x m_y b_x b_y)))
    (begin 
  (lengths_eq a_x a_y b_x b_y b_x b_y c_x c_y) 
  (displayln "This is a parallelogram.") 
  )
  (displayln "This is not a parallelogram.") 
))



(grobner_decide 0 0 1 2 4 2 3 0 2 1)
(grobner_decide_coll 0 0 1 2 4 2 3 0 2 1)
(grobner_decide_mid 0 0 1 2 4 2 3 0 2 1)