Extend sides

Line C'B'

Triangle DEF

Cevians

Ratios

Theorem (Ceva's): The lines $AA'$, $BB'$, $CC'$ are concurrent if and only if $\frac{AC'}{C'B}\frac{BA'}{A'C}\frac{CB'}{B'A}=1$

Ceva's Theorem

A = (-21.347062438482066, 17.175650098993778)
B = (-33.326878744043626, 4.52091456494991)
C = (-0.030863472248115715, 5.702023214794004)
poly1, c, a, b = Polygon(A, B, C)
d = Line(A, B)
e = Line(B, C)
f = Line(A, C)
C' = Point(d)
B' = Point(f)
A' = Point(e)
u = Vector(B, A')
v = Vector(B, C)
w = Vector(B, C')
z = Vector(B, A)
g = Vector(A, B')
h = Vector(A, C)
i = Line(A, A')
j = Line(B, B')
k = Line(C', C)
D = Intersect(i, j)
E = Intersect(j, k)
F = Intersect(i, k)
poly2, f_1, d_1, e_1 = Polygon(D, E, F)
l = Segment(C', C)
m = Segment(A, A')
n = Segment(B, B')
p = Line(C', B')
o = false
q = false
text4: "Area of $\Delta ABC= " + (LaTeX[poly1]) + ""
text5: "Area of $\Delta DEF=$ " + (LaTeX[poly2]) + ""
text6: "$\frac{\Delta DEF} {\Delta ABC}= $" + (LaTeX[poly2 / poly1]) + ""
r = false
text8: "\frac{AC'}{C'B}=" + (LaTeX[((z - w) * z / (z * z)) / ((w * z) / (z * z))]) + ""
text10: "\frac{CB'}{B'A}=" + (LaTeX[((h - g) * h / (h * h)) / ((g * h) / (h * h))]) + ""
text9: "\frac{BA'}{A'C}=" + (LaTeX[(u * v / (v * v)) / (((v - u) * v) / (v * v))]) + ""
text7: "\frac{AC'}{C'B}\frac{BA'}{A'C}\frac{CB'}{B'A}=" + (LaTeX[(u * v / (v * v)) / (((v - u) * v) / (v * v)) * ((z - w) * z / (z * z)) / ((w * z) / (z * z)) * ((h - g) * h / (h * h)) / ((g * h) / (h * h))]) + ""
a_1 = true
s = true
t = false