\; \text{Nájdite množinu všetkých bodov }X:\left|AA'  \right| =\left| BB' \right|  \text{, kde }A∈k,B∈l.

 \text{Postup} 

\text{Dané sú kružnice: }k(S_1,r_1),l(S_2,r_2); \;  \left| S_1S_2 \right| >r_1+r_2

1.  \; \text{Nech }X:\left|AA'  \right| =\left| BB' \right| ⇒AB∥A'B'⇒\red{\triangle ABX∼\triangle S_1S_2X}

2.  \; \frac{|XA|}{|XS_1|}=\frac{|XB|}{|XS_2|}, \text{ dosadením } |XA| = |XS_1| − r_1,|XB| = |XS_2| − r_2 

3.  \; \text{dostaneme }  \frac{|XS_1|}{|XS_2|}=\frac{r_1}{r_2}

4.  \; \text{ Pre } r_1=r_2 \text{ je to os úsečky bez jej stredu} \text{, pre } r_1≠r_2 \text{  Apolloniova kruž.

5.  \; \text{Musíme ešte ukázať: } X∈ω ⇒\left|AA'  \right| =\left| BB' \right| 

6.\; \text{Platnosť implikácie: ak platia vťahy v 3. tak platia aj vzťahy v 2. a 1. }

S_{2} = (4.5, 2.5)
B_{1} = (3.5, 1)
c = Circle(S_{2}, B_{1})
S_1 = (1.5, 2.5)
D = (1.7137751109200632, 1.7831906134382232)
d = Circle(S_1, D)
f = Line(S_1, S_{2})
A = Point(d)
g = OrthogonalLine(A, f)
A' = Intersect(f, g)
h = Ray(S_1, A)
i = OrthogonalLine(A, g)
B = Intersect(c, i, 2)
j = Ray(S_{2}, B)
X = Intersect(h, j)
GMB1 = Locus(X, A)
k = OrthogonalLine(B, f)
B' = Intersect(f, k)
l = Segment(A, A')
m = Segment(B, B')
t1, x_1, a, b = Polygon(A, B, X)
t2, x_2, s_1, s_{2} = Polygon(S_1, S_{2}, X)
GMB2 = Locus(X, A)