 \text{ Most cez rieku

\text{Posunutie: }τ_\hat{u}(A)=A'

$C$

C∈\overleftrightarrow{A'B} \text{ resp. }C_1∈\overleftrightarrow{A'B} 

$C_1$

$C^{-u}$

 \text{Dĺžka cesty bez mosta } 

$C_1^{u}$

$S$

$C_s$

$C'_s$

S∈o \; ∩ \; \overleftrightarrow{AB}

$C_2$

${C_2}^u$

$B^{-u} $

\text{K úplnosti je nutné dokázať, že body }A,C^{-u},B^{-u} \text{ sú kolineárne.

P = (0, 5)
f = Line(P, x = 0)
p = OrthogonalLine(P, f)
Q = Point(f)
q = OrthogonalLine(Q, f)
A = (0.722479338842978, 7.749917355371898)
B = (8.656363636363645, 1.7499173553719016)
u = Vector(P, Q)
d = Length(u)
A' = Translate(A, u)
v = Vector(A, A')
k = Line(A', B)
C = Intersect(q, k)
v_B = Distance(B, q)
v_A = Distance(A, p)
C_{1} = Intersect(p, k)
text6: "\text{Ak vzdialenosť }A\text{ od }p: \\
\text{ }\\
 \;  \;  \;  \;  \; d_p=  " + (LaTeX[v_A]) + "    \\ 
\text{ }\\
\text{ je väčšia alebo rovná ako vzdialenosť }B\text{ od }q:\\
\text{ }\\
\;  \;  \;  \;  \; d_q= " + (LaTeX[v_B]) + "    ,\\
\text{tak }\\
\text{ }\\
 \;  \;  \;  \;  \; C=k∩q,\\
\text{ }\\
 \text{ak nie, tak  }\\
\text{ }\\
\;  \;  \;  \;  \; C_1=k∩p,\\"
p_{1}: x(P)
j = OrthogonalLine(C, q)
C_{-u} = Intersect(p, j)
n = Segment(B, C)
r = Segment(C, C_{-u})
s = Segment(C_{-u}, A)
E = (10, 1)
F = (10, 0)
G = Midpoint(E, F)
d_z: n + s
text3: "\text{∙ \green {zelená:}  } \;  \;  \;  \green {  " + (LaTeX[d_z]) + "   }"
i_1 = OrthogonalLine(C_{1}, p)
C_{1u} = Intersect(i_1, q)
h_1 = Segment(A, C_{1})
j_1 = Segment(C_{1}, C_{1u})
k_1 = Segment(C_{1u}, B)
d_f: h_1 + k_1
text12: "\text{∙ \black {fialová}:  } \black { " + (LaTeX[d_f]) + " }   "
g = Line(A, B)
o = AngularBisector(p, q)
S = Intersect(o, g)
AF = Distance(A, C_{-u})
EB = Distance(C, B)
t = OrthogonalLine(S, o)
C_s = Intersect(p, t)
C'_s = Intersect(q, t)
a = Segment(A, C_s)
b = Segment(C_s, C'_s)
c = Segment(C'_s, B)
d_c: a + c
text5: "\text{∙ \red {červená}:  } \red {  " + (LaTeX[d_c]) + "  }"
C_2 = Point(p)
h = OrthogonalLine(C_2, p)
D = Intersect(h, q)
i = Segment(A, C_2)
l = Segment(C_2, D)
m = Segment(D, B)
d_m: i + m
text1: "\text{∙ \blue {modrá}:  } \blue { " + (LaTeX[d_m]) + "}"
w: (-u)
B' = Translate(B, w)
e = Vector(B, B')
f_1 = Segment(A, B')
t1, c_s, a_1, b' = Polygon(A, B', C_s)
text8: "\text{Podľa trojuholníkovej nerovnosti bude v  }\triangle AB'C_s:\\
\text{ }\\
 \;  \;  \;  \;  \;\green {" + (LaTeX[d_z]) + "}  =\left| AB' \right| <\left| B' C_s \right| +\left| AC_s \right| = \red {" + (LaTeX[d_c]) + " } "
t2, c_{1}, a_2, g_1 = Polygon(A, B', C_{1})
text7: "\text{Podľa trojuholníkovej nerovnosti bude v  }\triangle AB'C_1:\\
\text{ }\\
 \;  \;  \;  \;  \; " + (LaTeX[d_z]) + " =\left| AB' \right| <\left| B' C_1 \right| +\left| AC_1 \right| = " + (LaTeX[d_f]) + ""