$A$

$B$

$C$

$ Heron 公式 $
$\Delta =\sqrt{s(s-a)(s-b)(s-c)},s=\frac{1}{2}(a+b+c)$

$ ∵\Delta ABC=\frac{1}{2}\;ab\;\fgcolor{red}{sinC} $

$ =\frac{1}{2}\;ab\fgcolor{red}{\sqrt{1-cos^2C} }$

$ =\frac{1}{2}\;ab\;\fgcolor{red}{\sqrt{1-\left(  \frac{a^2+b^2-c^2}{2ab} \right)^2 } }$

$ =\frac{ab}{2}\cdot\sqrt{\frac{(2ab)^2-(a^2+b^2-c^2)^2}{(2ab)^2}} $

$ =\frac{ab}{2}\cdot\frac{1}{2ab}\cdot\sqrt{(2ab+a^2+b^2-c^2)(2ab-a^2-b^2+c^2)} $

$ =\frac{1}{4}\cdot \sqrt{[(a+b)^2-c^2][c^2-(a-b)^2]} $

$ =\frac{1}{4}\cdot \sqrt{(a+b+c)\fgcolor{red}{(a+b-c)}\fgcolor{green}{(c+a-b)}\fgcolor{0000FF}{(c-a+b)}} $

$ =\frac{1}{4}\cdot\sqrt{2s\fgcolor{red}{(2s-2c)}\fgcolor{green}{(2s-2b)}\fgcolor{0000FF}{(2s-2a)}} $

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

$Step=%v$

$ a $

$ b $

$ c $

A = (9.549062139419227, -3.9917623573752894)
c_1 = Circle(A, 3)
B = Point(c_1)
A' = Rotate(A, 240*°, B)
e = Ray(B, A')
d_1 = Circle(B, 5)
D = Intersect(d_1, e, 1)
c = Segment(A, B)
C: D
b = Segment(A, C)
a = Segment(C, B)
s_1: (a + b + c) / 2
e_1 = Circle(C, 7)
text10: "$ s=\frac{1}{2}\;(a+b+c)=\frac{1}{2}\;(" + (LaTeX[a]) + "+" + (LaTeX[b]) + "+" + (LaTeX[c]) + ") =" + (LaTeX[FractionText[s_1]]) + "$"
text11: "$ \Delta=\sqrt{" + (LaTeX[s_1]) + "\;(" + (LaTeX[s_1]) + "-" + (LaTeX[a]) + ")(" + (LaTeX[s_1]) + "-" + (LaTeX[b]) + ")(" + (LaTeX[s_1]) + "-" + (LaTeX[c]) + ")} $"
text12: "$ =\sqrt{" + (LaTeX[s_1]) + "\times" + (LaTeX[s_1 - a]) + "\times" + (LaTeX[s_1 - b]) + "\times" + (LaTeX[s_1 - c]) + "} $"
text13: "$ =" + (LaTeX[SurdText[sqrt(s_1 * (s_1 - a) * (s_1 - b) * (s_1 - c))]]) + " $"
s = 0 (slider from 0 to 13, step 1)
text17: "$ a=" + (LaTeX[a]) + " $"
text18: "$ b=" + (LaTeX[b]) + " $"
text19: "$ c=" + (LaTeX[c]) + " $"
poly1, c_2, a_1, b_1 = Polygon(A, B, C)