midLine1[r1_, r2_]:={(r1[[1]]+r2[[1]])/2,(r1[[2]]+r2[[2]])/2};
xpoint[r1_,r2_]:=midLine[r1,r2][[1]]+(r2[[2]]  - r1[[2]] )/(r2[[1]]-r1[[1]])*(midLine[r1,r2][[2]]);
xrad[r1_,r2_]:=((r1[[1]]-xpoint[r1,r2])^2+r1[[2]]^2)^(1/2);
chord1[r1_,r2_]:=ArcTan[r1[[2]] / ( r1[[1]]-xpoint[r1,r2] )];
chord2[r1_,r2_]:=ArcTan[r2[[2]] / ( r2[[1]]-xpoint[r1,r2] )];

Cir[r1_,r2_]:= Circle[{xpoint[r1,r2],0},xrad[r1,r2],{0,π}];

IP1[r1_,r2_] := {xpoint[r1,r2]+xrad[r1,r2],0} ;
IP2[r1_,r2_] := {xpoint[r1,r2]-xrad[r1,r2],0} ;
Abst[z1_,z2_]:=ArcCosh[1+ Abs[z1-z2]^2 / (2 *Im[z1]*Im[z2])];
Eucl[z1_,z2_]:=Abs[z1-z2];


PTC[p_]:=p[[1]]+I p[[2]];

PlotR = 1;

Manipulate[Graphics[{
Text[r1 , r1 + {0,+PlotR/12}],
Text[r2 , r2 + {0,+PlotR/12}], 
Text[IP1[r1,r2],IP1[r1,r2]+{0,- PlotR/12}],
Text[IP2[r1,r2],IP2[r1,r2]+{0,- PlotR/12}],
Text[N[Abst[PTC[r1],  PTC[r2]  ]], {0, -PlotR/6}],
Text["Distance", {-0.3, -PlotR/6}],
Text[N[Eucl[PTC[r1],  PTC[r2]  ]], {0, -0.2-PlotR/6}],
Text["Euclidean Distance", {-0.4, -PlotR/6-0.2}],
Dashing[{}], Line[{{-100,0},{100,0}}],  Cir[r1,r2],
PointSize[.015],Point@{r1,r2}},
PlotRange->2,ImageSize->800],
{{r1,{-1,1}},{-10,0},{10,10},Locator,Appearance->None},
{{r2,{1,1}},{-10,0},{10,10},Locator,Appearance->None},
SaveDefinitions->True]