clear all
; This is Heron's formula for the area
; of any triangle, given side lengths a, b, and c.

2s = a+b+c
area = (s*(s-a)*(s-b)*(s-c))^.5
eliminate s
pause
; This is how we arrive at Heron's formula for the area
; of any triangle, given side lengths a, b, and c, using the formula
; for the area of a trapezoid with side lengths a, b, c, and d,
; where a and c are the parallel sides (a is the longer parallel side).

; A trapezoid is a quadrilateral with
; two sides that are parallel to each other.

; Formula for the area of a trapezoid that is not a parallelogram:
trapezoid_area=(a+c)/(4*(a-c))*((a+b-c+d)*(a-b-c+d)*(a+b-c-d)*(-a+b+c+d))^.5
copy
replace c with 0 ; make the shorter parallel side length = 0
replace d with c ; Heron's formula:
pause
compare 2 ; simplify and compare with Heron's formula

; This is how we arrive at Heron's formula for the area
; of any triangle, given side lengths a, b, and c, using
; Brahmagupta's formula for the area of a cyclic quadrilateral,
; making one side length equal zero, to make a cyclic triangle.
; Since all triangles are cyclic (can be circumscribed by a circle),
; this gives the area for any triangle.

2s=a+b+c+d ; cyclic quadrilateral side lengths are a, b, c, and d
cyclic_area = ((s-a)*(s-b)*(s-c)*(s-d))^.5
eliminate s ; Brahmagupta's formula:
copy
replace d with 0 ; Heron's formula:
pause
compare 2 ; simplify and compare with Heron's formula