Le triomphe du chevron !

B' et C' sont les milieux de [AC] et [AB]
On a donc, par le lemme du chevron,
rouge= vert et vert= bleu, donc rouge= bleu
et, toujours par le chevron, mais dans l'autre sens,
on en déduit que G est sur la médiane (AA').

Preuve

A = (3.56, 4.38)
B = (-2.1, -3.78)
C = (5.7, -2.36)
poly1, c, a, b = Polygon(A, B, C)
A' = Midpoint(a)
B' = Midpoint(b)
C' = Midpoint(c)
d = Segment(A, A')
e = Segment(B, B')
f = Segment(C, C')
G = Intersect(d, e)
g = Segment(A, B)
h = Segment(B, C)
i = Segment(C, A)
poly2, b_1, a_1, g_1 = Polygon(A, G, B)
poly3, c_1, a_2, g_2 = Polygon(A, G, C)
poly4, c_2, g_3, b_2 = Polygon(G, B, C)
j = false
Curve(x) = NaN
k = Segment(G, A')

Tool: tgpin
Input: A, B, C, D
Output: h

a = Line(C, D)
b = Line(A, B)
E = Intersect(a, b)
F = Midpoint(A, E)
d = Circle(F, A)
f = OrthogonalLine(B, b)
G = Intersect(d, f, 2)
g = Circle(E, G)
H = Intersect(g, a, 2)
h = Circle(A, B, H)