海龍公式(Heron Foumula)
Author: 藍邦偉　老師

海龍(Heron)公式

Δ= \frac{ 1 }{ 2} absinC

= \sqrt{ s(s-a)(s-b)(s-c) } 

= \frac{ 1 }{4 } \sqrt{ (2s)(2s-2c)(2s-2b)(2s-2a) }  

= \frac{1}{2} ab \sqrt{ 1-cos²C } 

= \frac{ 1 }{ 2}  a b \sqrt{ 1-( \frac{a²+b²-c²  }{ 2ab} )² } 

= \frac{ 1 }{2 } ab \sqrt{  \frac{(2ab)²-(a²+b²-c²)²  }{(2ab)² } } 

= \frac{ 1 }{4 }  \sqrt{(2ab+a²+b²-c²)(2ab-a²-b²+c²) } 

= \frac{ 1 }{4 }  \sqrt{ [(a+b)²-c²][c²-(a-b)²] } 

= \frac{ 1 }{4 }  \sqrt{ (a+b+c)(a+b-c)(c+a-b)(c-a+b) } 

Δ= \sqrt{ s(s-a)(s-b)(s-c) },s= \frac{ 1 }{2 } (a+b+c) 

A = (8.651949910554562, 4.162754919499106)
B = (4, 0)
C = (10, 0)
多邊形1, c, a, b = Polygon(A, B, C)
d = Circle(A, B, C)
n = 4 (slider from 0 to 9, step 1)
α = Angle(B, A, C)
β = Angle(C, B, A)
γ = Angle(A, C, B)
r = Radius(d)
e = LineBisector(B, C)
f = LineBisector(A, C)
g = OrthogonalLine(A, a)
h = OrthogonalLine(B, b)
i = OrthogonalLine(C, c)