x0[τ_]=Sinh[τ];
x1[τ_]=Cosh[τ];
L=0.5;
τq=1;
q={x1[τq],x0[τq]}
p1=ParametricPlot[{x1[τ],x0[τ]},{τ,0,2},PlotRange->{{-0.2,2.0},{-0.2,1.5}},PlotStyle->{Red}];
p2=ParametricPlot[{x1[τ]-L,x0[τ]},{τ,0,2},PlotRange->{{-0.2,2.0},{-0.2,1.5}},PlotStyle->{Blue}];
p3=Graphics[{PointSize[0.02],Point[q]}];
p3a=Graphics[Text[Style[Q,Large],q+{0.1,0.01}]];
Show[{p1,p2,p3,p3a}]

Drawing the simultaneity line form Q

p4=ParametricPlot[q+t*q,{t,-1,0.3},PlotStyle->{Red,Dashed}]

Show[{p1,p2,p3,p3a,p4}]

Calculating intersection  of the simultaneity line from Q to the blue line

ns1=NSolve[q+t*q=={Cosh[τ]-L,Sinh[τ]},{t,τ}]
t1=t/.ns1[[2]]
τ1=τ/.ns1[[2]]
p={x1[τ1]-L,x0[τ1]};
p5=Graphics[{PointSize[0.02],Point[p]}];
p5a=Graphics[Text[Style[P,Large],p-{0.1,0.01}]];

Show[{p1,p2,p3,p3a,p4,p5,p5a}]

Repeating the same for another point.

τb=0.5;
b={x1[τb],x0[τb]};
p6=Graphics[{PointSize[0.02],Point[b]}];
p6a=Graphics[Text[Style[B,Large],b+{0.1,0.01}]];
p7=ParametricPlot[b+t*b,{t,-1,0.8},PlotStyle->{Red,Dashed}];
ns2=NSolve[b+t*b=={Cosh[τ]-L,Sinh[τ]},{t,τ}]
t2=t/.ns2[[2]]
τ2=τ/.ns2[[2]]
a={x1[τ2]-L,x0[τ2]};
p8=Graphics[{PointSize[0.02],Point[a]}];
p8a=Graphics[Text[Style[A,Large],a-{0.1,0.01}]];
Show[{p1,p2,p3,p3a,p4,p5,p5a,p6,p6a,p7,p8,p8a}]

Finding point C simultaneous with Q for the blue spaceship

ns3=NSolve[{-L,0}+t*(q-{-L,0})=={Cosh[τ]-L,Sinh[τ]},{t,τ}]

t3=t/.ns3[[2]]
τ3=τ/.ns3[[2]]

c={x1[τ3]-L,x0[τ3]};
p9=Graphics[{PointSize[0.02],Point[c]}];
p9a=Graphics[Text[Style[C,Large],c-{0.1,0.01}]];
Show[{p1,p2,p3,p3a,p4,p5,p5a,p6,p6a,p7,p8,p8a,p9,p9a}]

Drawing simmultaneity line for C

p10=ParametricPlot[{-L,0}+t*(q-{-L,0}),{t,0,1.3},PlotStyle->{Blue,Dashed}]

obrazek=Show[{p1,p2,p3,p3a,p4,p5,p5a,p6,p6a,p7,p8,p8a,p9,p9a,p10}]

Calculating distances

d[x_,y_]:=Sqrt[(x[[1]]-y[[1]])^2-(x[[2]]-y[[2]])^2]

dab=d[a,b]

dpq=d[p,q]

dcq=d[c,q]

Export["E:/Mytex/dwiehiperbole.jpg",obrazek,ImageSize->570]

N[q]

{1.5430806348152437`,1.1752011936438014`}
N[b]

N[p]

N[a]

Sqrt[(0.943873^2-0.718848^2)]

q-p