ClearAll["Global`*"]

c={3*(1-t^2),2*(2*t)}/(1+t^2)
dc=Factor[D[c,t]]
ddc=Factor[D[dc,t]]

rych=Simplify[Sqrt[Factor[dc.dc]], t>0]

kri=Factor[Det[{dc,ddc}]/rych^3]

Plot[kri, {t,-10,10}, AxesOrigin->{0,0}]

pl1=ParametricPlot[c,{t,-10,10}, PlotRange->All, PlotStyle->Blue];

t0=2/5

c0=c/.{t->t0}

pl2=Graphics[{Blue, PointSize[0.013],Point[c0]}];

kri0=kri/.{t->t0}

n={{0,-1},{1,0}}.(dc/rych)

R=1/kri

S=Factor[c+n*R]

R0=R/.{t->t0}
S0=S/.{t->t0}

okru0=S0+R0{1-s^2,2s}/(1+s^2)

pl3=ParametricPlot[okru0, {s,-10, 10}, PlotStyle->{Dashed, Magenta}];

pl4=Graphics[{Magenta, PointSize[0.013],Point[S0]}];

Show[pl1, pl2, pl3, pl4]

n

nP=Factor[c+n*s]

nP0=nP/.{t->t0}

pl5=ParametricPlot[nP0, {s,0,5}, PlotStyle->{Thin, Black}];

Show[pl5, pl1, pl2, pl3, pl4, PlotRange->All]

pl6[T_]:=ParametricPlot[nP/.{t->T}, {s,0,5}, PlotStyle->{Thin, Black}];

res=Solve[(nP/.{s->r})==nP0,{r,s}][[1]]

prus=Factor[nP0/.res]

pl7[T_]:=Graphics[{Black, PointSize[0.013],Point[prus/.{t->T}]}]

Manipulate[Show[pl5, pl1, pl2, pl3, pl4, pl6[T],pl7[T],  PlotRange->All], {{T, t0}, 0,1}]