$ 投影定理$

$ \left\lbrace \begin{array}{ll} a=b\;cosC+c\;cosB \ b=a\;cosC+c\;cosA \ c=a\;cosB+b\;cosA \end{array}  \right.\  $

$ b\;cosC+c\;cosB $

$ =b\cdot\frac{a^2+b^2-c^2}{2ab}+c\cdot\frac{a^2+c^2-b^2}{2ac} $

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

$ =a $

$B$

$C$

$A$

$H$

$ b\;cosC+c\;cosB $

F = (11.04, 3.4)
E = (5.5, -0.72)
D = (6.58, 4.74)
s = 0 (slider from 0 to 9, step 1)
B = DynamicCoordinates(E, floor(x(E)), floor(y(E)))
C = DynamicCoordinates(F, floor(x(F)), floor(y(F)))
L = Line(B, C)
a = Segment(B, C)
A = DynamicCoordinates(D, floor(x(D)), floor(y(D)))
b = Segment(C, A)
c = Segment(A, B)
d = OrthogonalLine(A, L)
H = Intersect(d, L)
h = Segment(A, H)
α = Angle(B, A, C)
β = Angle(A, B, C)
γ = Angle(A, C, B)
poly1, c_1, a_1, b_1 = Polygon(A, B, C)
δ = Angle(B, H, A)
text6: "$= " + (LaTeX[b]) + "\;cos\;" + (LaTeX[γ]) + "+" + (LaTeX[c]) + "\;cos\;" + (LaTeX[β]) + " $"
text8: "$ =" + (LaTeX[SurdText[b * cos(γ)]]) + " " + (LaTeX[If[c * cos(β) < 0, "-", "+"]]) + "" + (LaTeX[SurdText[c * cos(β)]]) + "$"
text9: "$ =" + (LaTeX[SurdText[a]]) + "=a $"
text7: "$ =" + (LaTeX[b]) + "\times" + (LaTeX[SurdText[cos(γ)]]) + "+" + (LaTeX[c]) + " \times" + (LaTeX[SurdText[cos(β)]]) + "$"