$B$

$C$

$A$

$ 三角形的中線長公式 $

$ \left\lbrace \begin{array}{ll} m_a=\frac{1}{2}\sqrt{2(b^2+c^2)-a^2} \ m_b=\frac{1}{2}\sqrt{2(a^2+c^2)-b^2} \ m_c=\frac{1}{2}\sqrt{2(a^2+b^2)-c^2} \end{array}  \right.\ $

$M$

$ 設∠AMB=θ,∠AMC=180°-θ $

$ \Delta ABM中：\;$
$c^2=m_a^2+(\frac{a}{2})^2-2\;m_a\cdot\frac{a}{2}cosθ $
$\;\;\;\;=m_a^2+\frac{a^2}{4}-a\;m_a\;cosθ$

$ \Delta ACM中：\;$
$b^2=m_a^2+(\frac{a}{2})^2-2\;m_a\cdot \frac{a}{2}cos(180°-θ) $
$\;\;\;\;=m_a^2+\frac{a^2}{4}+a\;m_acosθ$

$ ⇒b^2+c^2=2m_a^2+\frac{a^2}{2} $
$⇒m_a^2=\frac{2(b^2+c^2)-a^2}{4}$

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

$  θ$

$ 180°-θ $

$ b $

$ c $

$ \frac{a}{2} $

$ \frac{a}{2} $

F = (10.08, -0.94)
E = (4.08, -0.64)
D = (8.56, 3.86)
B = DynamicCoordinates(E, floor(x(E)), floor(y(E)))
C = DynamicCoordinates(F, floor(x(F)), floor(y(F)))
a = Segment(B, C)
A = DynamicCoordinates(D, floor(x(D)), floor(y(D)))
b = Segment(C, A)
c = Segment(A, B)
M = Midpoint(B, C)
d = Segment(A, M)
s = 4 (slider from 0 to 8, step 1)
text7: "$ =\frac{1}{2}\times" + (LaTeX[SurdText[sqrt(2 * b^(2) + 2 * c^(2) - a^(2))]]) + " $"
text8: "$ =" + (LaTeX[SurdText[0.5 * sqrt(2 * b^(2) + 2 * c^(2) - a^(2))]]) + " $"
poly1, b_1, a_1, m = Polygon(A, M, B)
poly2, c_1, a_2, m_1 = Polygon(A, M, C)
text11: "$ b=" + (LaTeX[b]) + " $"
text12: "$ c=" + (LaTeX[c]) + " $"
text13: "$ \frac{a}{2} =" + (LaTeX[a / 2]) + "$"
text14: "$ \frac{a}{2} =" + (LaTeX[a / 2]) + "$"
text15: "$ m_a=" + (LaTeX[SurdText[d]]) + " $"
text6: "$ ⇒m_a=\frac{1}{2}\sqrt{2(b^2+c^2-a^2)}$
$\;\;\;\;\;\;\;\;\;\;=\frac{1}{2}\sqrt{2(" + (LaTeX[b]) + "^2+" + (LaTeX[c]) + "^2)-" + (LaTeX[a]) + "^2} $"