Midpoint of a segment From SICP:


  Exercise 2.2
  
  Consider the problem of
  representing line segments in a plane.
  Each segment is represented as a pair
  of points: a starting point and an
  ending point. Define a constructor
  make-segment and selectors
  start-segment and end-segment that
  define the representation of segments
  in terms of points. Furthermore, a
  point can be represented as a pair of
  numbers: the x coordinate and the y
  coordinate. Accordingly, specify a
  constructor make-point and selectors
  x-point and y-point that define this
  representation. Finally, using your
  selectors and constructors, define a
  procedure midpoint-segment that takes
  a line segment as argument and returns
  its midpoint (the point whose
  coordinates are the average of the
  coordinates of the endpoints). To try
  your procedures, you'll need a way to
  print points:

(define (print-point p)
  (newline)
  (display "(")
  (display (x-point p))
  (display ",")
  (display (y-point p))
  (display ")"))



I wrote the following solution:

(define (make-segment a b) (cons a b))
(define (start-segment l) (car l))
(define (end-segment l) (cdr l))

(define (make-point x y) (cons x y))
(define (x-point p) (car p))
(define (y-point p) (cdr p))

(define (sum . l) (if (zero? (length l)) 0 (+ (car l) (apply sum (cdr l)))))
(define (average . l) (/ (apply sum l) (length l)))

(define (midpoint seg)
  (make-point (average (x-point (start-segment seg)) 
                       (x-point (end-segment seg)))
              (average (y-point (start-segment seg))
                       (y-point (end-segment seg)))))

(define (print-point p)
  (newline)
  (display "(")
  (display (x-point p))
  (display ",")
  (display (y-point p))
  (display ")"))

(define seg-1 (make-segment (make-point 3 4)
                            (make-point 8 10)))
(print-point (midpoint seg-1))


What do you think?
Your definitions of make-segment and make-point can be simply bound to cons; you may do the same for accessors.

(define make-segment cons)
(define start-segment car)
(define end-segment cdr)

(define make-point cons)
(define x-point car)
(define y-point cdr)


Your definition of sum is similar to how + works.  You may use + instead in your definition of average.

(define (average . l) (/ (apply + l) (length l)))


Notice the repetition of code in your definition of midpoint--you create a new point out of the averages of each coordinate of the two end-points that describe the line segment.  You may abstract out the calculation of averages either by writing your own helper function or by using the built-in map function, like so:

(define point-&gt;coordinate-accessors (list x-point y-point))
(define (midpoint seg)
  (apply make-point (map (lambda (point-&gt;coordinate)
                           (average (point-&gt;coordinate (start-segment seg))
                                    (point-&gt;coordinate (end-segment seg))))
                         point-&gt;coordinate-accessors)))


This definition is general enough to handle midpoint calculation in as many dimensions as handled by make-point and point-&gt;coordinate-accessors.