solution

ABC isocèle en A, la médiatrice de [AC] coupe (BC)
en D. On porte E tel que AE= BD. 
Montrer que DCE est isocèle en C.

Transformations 1

Concours 1

angles

Transformations 2

Concours 2

E' est le symétrique de E par rapport à (DH)

E'' est le symétrique de D par rapport à (AK)

A = (4.86898700875161, 8.665312068906049)
B = (-1.8868073059274544, -9.8648666227851)
c = Circle(A, B)
C = Point(c)
t1, c_1, a, b = Polygon(A, B, C)
f = Line(B, C)
g = LineBisector(b)
D = Intersect(f, g)
h = Line(D, A)
d = Circle(A, Segment[D, B])
E = Intersect(d, h, 2)
i = Segment(C, E)
j = Segment(D, B)
k = Segment(A, E)
H = Intersect(g, b)
α = Angle(D, H, C)
t2, d_1, a_1, b_1 = Polygon(A, B, D)
t3, e, a_2, c_2 = Polygon(A, C, E)
l = false
n = OrthogonalLine(A, a)
K = Intersect(f, n)
β = Angle(C, K, A)
E' = Mirror(E, g)
m = false
O = Intersect(g, n)
p = Line(C, O)
G = Intersect(h, p)
q = Segment(E, E')
o = false
γ = Angle(C, D, A)
δ = Angle(D, E, C)
ε = Angle(C, B, A)
ζ = Angle(A, C, B)
η = Angle(D, A, C)
r = false
s = false
E'' = Intersect(f, q)
t = false
θ = Angle(D, G, C)
f_1 = Segment(C, E'')
g_1 = Segment(A, E')

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)