Tesi\!:\ la\ bisettrice\ di\ un\ angolo\ esiste\
ed\ è\ unica

1\ -\ \overline{OA} \cong \overline{OC};\ 
\overline{AB} \cong \overline{CD}\ -\ Per\ costruzione

2\ -\ \overline{OB} \cong \overline{OD}\ -\ 
Per\ (1)

3\ -\ \triangle AOD \cong \triangle COB\ -\ 
I\ crit.\!: \overline{OA} \cong \overline{OC}\ (1);\ 
\overline{OB} \cong \overline{OD}\ (2);\ 
A\hat{O}C\ comune

4\ -\ O\hat{C}B \cong O\hat{A}D;\ 
O\hat{B}C \cong O\hat{D}A\ -\ Per\ (3)

5\ -\ B\hat{C}D \cong D\hat{A}B\ -\ 
Supplementari\ di\ angoli\ congruenti (4)

6\ -\ \triangle AEB \cong \triangle CED\ -\ 
II\ crit.\!:\ \overline{AB} \cong \overline{CD}\ 
(1);\ E\^CD \cong E\^AB\ (5);\ 
\ \ \ \ \ \ \ O\^BE \cong O\^DE\ (4)

7\ -\ \overline{AE} \cong \overline{CE}\ -\ 
Per\ (6)

8\ -\ \triangle AOE \cong \triangle OCE\ -\ 
I\ crit.\!:\ \overline{OA} \cong \overline{OC}\ 
(1);\ \overline{AE} \cong \overline{CE}\ 
(7);\ O\^CE \cong O\^AE\ (4)

9\ -\ A\hat{O}E \cong C\hat{O}E\ -\ Per\ (8)\ -\ La\ bisettrice\ esiste

10\ -\  Per\ assurdo\ sia\ OK\ bisettrice\ con\ K 
\notin OE\ -\ Negazione\ della\ tesi

11\ -\ K\hat{O}A \cong \frac{1}{2} C\hat{O}A
\cong E\hat{O}A\ -\ Proprietà\ della\ bisettrice\ 
-\ Impossibile\
\ \ \ \ \ \ \ \ \ perchè\ E\ interno\ \ K\hat{O}A

12\ -\ La\ bisettrice\ è\ unica\ - (10)\ è\ falsa\ 
poiché\ (11)\ è\ impossibile\
Q.\ E.\ D.

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O = (0.02, 3.54)
P = (-1.84, -0.64)
a = Ray(O, P)
Q = (2, -0.32)
b = Ray(O, Q)
A = Point(a)
c = Circle(O, A)
B = Point(a)
d = Circle(O, B)
e = Segment(O, A)
f = Segment(A, B)
C = Intersect(c, b, 1)
g = Segment(O, C)
D = Intersect(d, b, 1)
h = Segment(C, D)
i = Segment(A, D)
j = Segment(B, C)
E = Intersect(i, j)
k = Ray(O, E)
poli1, a_1, o, d_1 = Polygon(O, D, A)
poli2, c_1, o_1, b_1 = Polygon(O, B, C)
poli3, e_1, d_2, c_2 = Polygon(D, C, E)
poli4, e_2, a_2, b_2 = Polygon(A, B, E)
poli5, e_3, o_2, c_3 = Polygon(O, C, E)
poli6, e_4, a_3, o_3 = Polygon(A, O, E)
K = (0.96, -1.6)
l = Ray(O, K)