Lösungen

Numerisch

{3/7,6/7,9/7}//N

{6/7,12/7,18/7}//N

6-36 t/7 //Together

{6/7(7-6t),12/7,18/7}//N //Expand

{5/14,19/7,29/14}//N

{5/7,24/7,22/7}//N

{31/7,13/7,9/7}//N

{34/7,19/7,18/7}//N

{5/7+29 t/7,24/7-5t/7,22/7-4t/7}//N

{0,103/29,94/29}//N

{154/69,-59/69,37/69}//N

{170/69,-187/69,5/69}//N

{80/23,19/23,24/23}//N

{91/23,107/23,2/23}//N

{170/69+103 t/69, 187/69-134t/69,5/69+t/69}//N

{0,51/103,5/103}//N

{5/6,5/6,2/3}//N

{2/3,2/3,5/6}//N

{3/4,3/4,1/2}//N

{2/3+t/12,2/3+t/12,5/6-t/3}//N

{7/8,7/8,0}//N

Skizzen

Dreiersummen von Quadratzahlen

tabT1=Flatten[Table[Sqrt[n^2+m^2+k^2],{n,1,10},{m,1,10},{k,1,10}]];

tabT1a=Table[{j,IntegerQ[Flatten[Table[Sqrt[n^2+m^2+k^2],{m,1,10},{n,1,10},{k,1,10}]][[j]]]},{j,1,Length[tabT]}];

tabT2=Drop[Union[Table[If[IntegerQ[Flatten[Table[Sqrt[n^2+m^2+k^2],{m,1,10},{n,1,10},{k,1,10}]][[j]]],j,0],{j,1,Length[tabT1]}]],1];

tabT2=Drop[Union[Table[If[IntegerQ[Evaluate[tabT1[[j]]]],j,0],{j,1,Length[tabT1]}]],1];

tabT2a={12,38,74,102,111,126,134,153,159,186,216,252,256,308,314,332,337,364,371,378,500,513,519,522,526,553,557,566,582,634,656,704,731,73,774,816,852,950,995};

tabT3=Flatten[Table[{m,n,k},{m,1,10},{n,1,10},{k,1,10}],2];

Table[tabT3[[tabT2a[[j]]]],{j,1,Length[tabT2a]}]

Remove["Global`*"]

Benötigt vorgängigen Run von "GeoSourceCode.nb".
Zur Benutzung: In den Beispielen können die Eingabewerte überschrieben werden.

lotFussPunktEbene[pG1_,pG2_,pG3_,pQ_]:=
    Module[
      {ebene,λ,μ,n,g,solv,pL1,pL2},
      
      Off[General::spell];
      ebene[λ_,μ_]:=pG1+ λ (pG2-pG1)+μ(pG3-pG1);
      (*  *) 
      If[var1Stop2Never3Use==9,Print[ebene[λ99,μ99]] ,
        leer=leer;];
      n=Cross[(pG2-pG1),(pG3-pG1)];
      (*  *) If[var1Stop2Never3Use==9,Print[n] ,leer=leer;];
      g[ν_]:=pQ+ν n;
      (*  *) If[var1Stop2Never3Use==9,Print[g[ν99]] ,leer=leer;];
      solv=Flatten[
          Solve[ebene[λ,μ]==g[ν],{λ,μ,ν}]];
      (*  *) If[var1Stop2Never3Use==9,Print[solv] ,leer=leer;];
      pL1=ebene[λ,μ]/.{solv[[1]],solv[[2]]};
      (*  *) If[var1Stop2Never3Use==9,Print[pL1] ,leer=leer;];
      pL2=g[ν]/.{solv[[3]]};(*  *) 
      If[var1Stop2Never3Use==9,Print[pL2] ,leer=leer;];
      Print["Lotfusspunkt =  ",pL1," = ",pL1 //N];
      (*  *) 
      If[var1Stop2Never3Use==9,Print["Exakt: ",pL2] ,leer=leer;];
     pOut=pL1 ];

pU1={12,0,0};
pU2={0,6,0};
pU3={0,0,4};
pU4={2,0,0};
pU5={0,2,1};
nu={0,0,0};
pU6=nu;
pU7={0,2,1};
pU8={2,1,0};
pa1={1,0,0};
pa2={0,1,0};
pa3={0,0,1};
pS4=lotFussPunktEbene[pU1,pU2,pU3,pU4];
pS5=lotFussPunktEbene[pU1,pU2,pU3,pU5];
pS6=lotFussPunktEbene[pU1,pU2,pU3,nu];
pS7=lotFussPunktEbene[pU1,pU2,pU3,pU7];
pS8=lotFussPunktEbene[pU1,pU2,pU3,pU8];
Print[pS6];
Print[pS7];
Print[pS8];
pA6=pU6+2(pS6-pU6);
pA7=pU7+2(pS7-pU7);
pA8=pU8+2(pS8-pU8);
pN4=pU4+2(pS4-pU4);
pN5=pU5+2(pS5-pU5);

po3[p__,size_]:=Show[Graphics3D[{PointSize[size],Map[Point,{p}]}],PlotRange->All,AspectRatio->Automatic,DisplayFunction->Identity];
po3[{p__},size_]:=po3[p,size];

kreuz=Show[Graphics3D[{Line[{nu-pa1,pU1+pa1}],Line[{nu-pa2,pU2+pa2}],Line[{nu-pa3,pU3+pa3}]}],PlotRange->All,AspectRatio->Automatic,DisplayFunction->Identity];

l3[p__,d_]:=Show[Graphics3D[{Thickness[d],Line[{p}]}],PlotRange->All,
      AspectRatio->Automatic,DisplayFunction->Identity];
l3[{p__,d_}]:=l3[p,d];

s1=Show[po3[{pU1,pU2,pU3},0.04],kreuz,DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s1a=Show[po3[{pU1,pU2,pU3},0.04],kreuz,DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];

s2b=Show[kreuz,l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic];
s2=Show[kreuz,l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s2a=Show[kreuz,l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];
(* s3=Show[po3[{pU1,pU2,pU3,pU4,pU5,pS4,pS5,pN4,pN5},0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic];
*)
s5a=Show[kreuz,po3[{nu,pS6,2pS6},0.04],
l3[nu,pS6,2pS6,0.01], l3[pU1,pU2,pU3,pU1,0.02],l3[nu,pU1,2pS6,0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s5b=Show[kreuz,po3[{nu,pS6,2pS6},0.04],
l3[nu,pS6,2pS6,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[nu,pU1,2pS6,0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];
s5c=Show[kreuz,po3[{pU7,pS7,pA7,pU8,pS8,pA8},0.04],
l3[pU7,pU8,pA8,pA7,pU7,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU7,pU8+1.5(pU8-pU7),0.01],l3[pA7,pA8+1.5(pA8-pA7),0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s5d=Show[kreuz,po3[{pU7,pS7,pA7,pU8,pS8,pA8},0.04],
l3[pU7,pU8,pA8,pA7,pU7,0.01],l3[pU7,pU8+1.5(pU8-pU7),0.01],l3[pA7,pA8+1.5(pA8-pA7),0.01],
l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];




s5=Show[kreuz,l3[pU4,pN4,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pU3,0.01],po3[{pU1,pU2,pU3,pU4 ,pS4, pN4 },0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s5=Show[kreuz,l3[pU4,pN4,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pU3,0.01],po3[{pU1,pU2,pU3,pU4 ,pS4, pN4 },0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,ViewPoint->{3.630, 2.5, 3},Boxed->False];
s5=Show[kreuz,l3[pU4,pN4,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pU3,0.01],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,0.02],po3[pU4 ,pS4, pN4,0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,ViewPoint->{-1.330, -5.079, 1.030},Boxed->False];

s4=Show[kreuz,po3[{pU4,pU5},0.04],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU5,pU2,pU5,pU3,0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s4=Show[kreuz,po3[{pU4,pU5},0.04],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU5,pU2,pU5,pU3,0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,ViewPoint->{3.630, 2.5, 3},Boxed->False];

s6=Show[kreuz,l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU5,pU2,pU5,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pN5,pU2,pN5,pU3,0.01],po3[{pU5, pN5,pU4, pN4},0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];

Print[pS6];
Print[pS7];
Print[pS8];
Print[pA6];
Print[pA7];
Print[pA8];

Show[s3,s4,s2,kreuz];
Show[s3,s4,s2,kreuz,Boxed->False];
Show[s3,s2,s4,kreuz,Boxed->False,ViewPoint->{3.630, 2.5, 3}];
s5=Show[l3[pU4,pN4,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pU3,0.01],po3[{pU1,pU2,pU3,pU4 ,pS4, pN4 },0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,ViewPoint->{3.630, 2.5, 3}];

s6=Show[l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU5,pU2,pU5,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pN5,pU2,pN5,pU3,0.01],po3[{pU5, pN5,pU4, pN4},0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,ViewPoint->{3.630, 2.5, 3}];

kreuz=Show[Graphics3D[{Line[{nu-pa1,pU1+pa1}],Line[{nu-pa2,pU2+pa2}],Line[{nu-pa3,pU3+pa3}]}],PlotRange->All,AspectRatio->Automatic];

s5=Show[kreuz,l3[pU4,pN4,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,pN4,pU1,pN4,pU2,pN4,pU3,pN4,pU3,0.01],l3[pU4,pU1,pU4,pU2,pU4,pU3,pU4,pU3,0.02],po3[pU4 ,pS4, pN4,0.04],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,ViewPoint->{-1.330, -5.079, 1.030},Boxed->False];

Options[Graphics3D]

2y+3y+6z==24

Solve[12 a==1]

lotFussPunktEbene[pG1_,pG2_,pG3_,pQ_]:=
    Module[
      {ebene,λ,μ,n,g,solv,pL1,pL2},
      
      Off[General::spell];
      ebene[λ_,μ_]:=pG1+ λ (pG2-pG1)+μ(pG3-pG1);
      (*  *) 
      If[var1Stop2Never3Use==9,Print[ebene[λ99,μ99]] ,
        leer=leer;];
      n=Cross[(pG2-pG1),(pG3-pG1)];
      (*  *) If[var1Stop2Never3Use==9,Print[n] ,leer=leer;];
      g[ν_]:=pQ+ν n;
      (*  *) If[var1Stop2Never3Use==9,Print[g[ν99]] ,leer=leer;];
      solv=Flatten[
          Solve[ebene[λ,μ]==g[ν],{λ,μ,ν}]];
      (*  *) If[var1Stop2Never3Use==9,Print[solv] ,leer=leer;];
      pL1=ebene[λ,μ]/.{solv[[1]],solv[[2]]};
      (*  *) If[var1Stop2Never3Use==9,Print[pL1] ,leer=leer;];
      pL2=g[ν]/.{solv[[3]]};(*  *) 
      If[var1Stop2Never3Use==9,Print[pL2] ,leer=leer;];
      Print["Lotfusspunkt =  ",pL1," = ",pL1 //N];
      (*  *) 
      If[var1Stop2Never3Use==9,Print["Exakt: ",pL2] ,leer=leer;];
     pOut=pL1 ];
pU1={12,0,0};
pU2={0,8,0};
pU3={0,0,4};
pU4={2,0,0};
pU5={0,2,1};
nu={0,0,0};
pU6=nu;
pU7={0,2,1};
pU8={4,1,0};
pa1={1,0,0};
pa2={0,1,0};
pa3={0,0,1};
pS4=lotFussPunktEbene[pU1,pU2,pU3,pU4];
pS5=lotFussPunktEbene[pU1,pU2,pU3,pU5];
pS6=lotFussPunktEbene[pU1,pU2,pU3,nu];
pS7=lotFussPunktEbene[pU1,pU2,pU3,pU7];
pS8=lotFussPunktEbene[pU1,pU2,pU3,pU8];
pA6=pU6+2(pS6-pU6);
pA7=pU7+2(pS7-pU7);
pA8=pU8+2(pS8-pU8);
pN4=pU4+2(pS4-pU4);
pN5=pU5+2(pS5-pU5);

po3[p__,size_]:=Show[Graphics3D[{PointSize[size],Map[Point,{p}]}],PlotRange->All,AspectRatio->Automatic,DisplayFunction->Identity];
po3[{p__},size_]:=po3[p,size];

kreuz=Show[Graphics3D[{Line[{nu-pa1,pU1+pa1}],Line[{nu-pa2,pU2+pa2}],Line[{nu-pa3,pU3+pa3}]}],PlotRange->All,AspectRatio->Automatic,DisplayFunction->Identity];

l3[p__,d_]:=Show[Graphics3D[{Thickness[d],Line[{p}]}],PlotRange->All,
      AspectRatio->Automatic,DisplayFunction->Identity];
l3[{p__,d_}]:=l3[p,d];

s1=Show[po3[{pU1,pU2,pU3},0.04],kreuz,DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s1a=Show[po3[{pU1,pU2,pU3},0.04],kreuz,DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];

s2=Show[kreuz,l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False];
s2a=Show[kreuz,l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];

s5a=Show[kreuz,po3[{nu,pS6,2pS6},0.04],
l3[nu,pS6,2pS6,0.01], l3[pU1,pU2,pU3,pU1,0.02],l3[nu,pU1,2pS6,0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{1.3,-2.6,2.}];

s5b=Show[kreuz,po3[{nu,pS6,2pS6},0.04],
l3[nu,pS6,2pS6,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[nu,pU1,2pS6,0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];

s5c=Show[kreuz,po3[{pU7,pS7,pA7,pU8,pS8,pA8},0.04],
l3[pU7,pU8,pA8,pA7,pU7,0.01],l3[pU1,pU2,pU3,pU1,0.02],l3[pU7,pU8+1.5(pU8-pU7),0.01],l3[pA7,pA8+1.5(pA8-pA7),0.01],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{1.3,-2.6,2.}];

s5d=Show[kreuz,po3[{pU7,pS7,pA7,pU8,pS8,pA8},0.04],
l3[pU7,pU8,pA8,pA7,pU7,0.01],l3[pU7,pU8+1.5(pU8-pU7),0.01],l3[pA7,pA8+1.5(pA8-pA7),0.01],
l3[pU1,pU2,pU3,pU1,0.02],DisplayFunction->$DisplayFunction,PlotRange->All,AspectRatio->Automatic,Boxed->False,ViewPoint->{3.630, 2.5, 2}];

{1.3,-2.4,2.} ViewPoint->{1.3,-2.4,2.}

Print[pS6];
Print[pS7];
Print[pS8];
Print[pA6];
Print[pA7];
Print[pA8];

lotFussPunktEbene[pG1_,pG2_,pG3_,pQ_]:=
    Module[
      {ebene,λ,μ,n,g,solv,pL1,pL2},
      
      Off[General::spell];
      ebene[λ_,μ_]:=pG1+ λ (pG2-pG1)+μ(pG3-pG1);
      (*  *) 
      If[var1Stop2Never3Use==9,Print[ebene[λ99,μ99]] ,
        leer=leer;];
      n=Cross[(pG2-pG1),(pG3-pG1)];
      (*  *) If[var1Stop2Never3Use==9,Print[n] ,leer=leer;];
      g[ν_]:=pQ+ν n;
      (*  *) If[var1Stop2Never3Use==9,Print[g[ν99]] ,leer=leer;];
      solv=Flatten[
          Solve[ebene[λ,μ]==g[ν],{λ,μ,ν}]];
      (*  *) If[var1Stop2Never3Use==9,Print[solv] ,leer=leer;];
      pL1=ebene[λ,μ]/.{solv[[1]],solv[[2]]};
      (*  *) If[var1Stop2Never3Use==9,Print[pL1] ,leer=leer;];
      pL2=g[ν]/.{solv[[3]]};(*  *) 
      If[var1Stop2Never3Use==9,Print[pL2] ,leer=leer;];
      Print["Lotfusspunkt =  ",pL1," = ",pL1 //N];
      (*  *) 
      If[var1Stop2Never3Use==9,Print["Exakt: ",pL2] ,leer=leer;];
     pOut=pL1 ];
pU1={1,0,0};
pU2={0,1,0};
pU3={1,1,1};
pU4={2,0,0};
pU5={0,2,1};
nu={0,0,0};
pU6=nu;
pU7={1,1,1/2};
pU8={0,0,1/2};
pa1={1,0,0};
pa2={0,1,0};
pa3={0,0,1};
pS4=lotFussPunktEbene[pU1,pU2,pU3,pU4];
pS5=lotFussPunktEbene[pU1,pU2,pU3,pU5];
pS6=lotFussPunktEbene[pU1,pU2,pU3,nu];
pS7=lotFussPunktEbene[pU1,pU2,pU3,pU7];
pS8=lotFussPunktEbene[pU1,pU2,pU3,pU8];
pA6=pU6+2(pS6-pU6);
pA7=pU7+2(pS7-pU7);
pA8=pU8+2(pS8-pU8);
pN4=pU4+2(pS4-pU4);
pN5=pU5+2(pS5-pU5);
(*
 {6,0,0}+t(2 lotFussPunktEbene[pU1,pU2,pU3,nu]-{6,0,0})//TeXForm ;
 
Solve[Evaluate[(lotFussPunktEbene[pU1,pU2,pU3,pU8]+t (lotFussPunktEbene[pU1,pU2,pU3,pU7]-lotFussPunktEbene[pU1,pU2,pU3,pU8]))[[3]]]==0,{t}];
lotFussPunktEbene[pU1,pU2,pU3,pU7]//TeXForm ;
lotFussPunktEbene[pU1,pU2,pU3,pU8]//TeXForm ;

pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7) 
pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7) //TeXForm 
pU8+2(lotFussPunktEbene[pU1,pU2,pU3,pU8]-pU8) //TeXForm   
Solve[(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * (pU8+2(lotFussPunktEbene[pU1,pU2,pU3,pU8]-pU8)-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7))))[[1]]==0,{t}];
 (pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * (pU8+2(lotFussPunktEbene[pU1,pU2,pU3,pU8]-pU8)-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)))) //TeXForm 
 (pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * (pU8+2(lotFussPunktEbene[pU1,pU2,pU3,pU8]-pU8)-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)))) /. t -> 5/29 ;
 (pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * (pU8+2(lotFussPunktEbene[pU1,pU2,pU3,pU8]-pU8)-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)))) /. t -> -5/29  //TeXForm   ;
 Solve[ (pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * ( sch-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7))))[[1]]==0,{t}];
 (pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * ( sch-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7))))/.t->-8
  *)

 (pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7)+t * ( sch-(pU7+2(lotFussPunktEbene[pU1,pU2,pU3,pU7]-pU7))))/.t->5/2 //TeXForm

schnitt[ps1_,ps2_,ps3_,qs1_,qs2_]:=
    Module[{ebene,gerade,λ,μ,t,solv,solv1,sys},
      Off[General::spell];
      ebene[λ_,μ_]:=ps1+λ(ps2-ps1)+μ(ps3-ps1);(* *) 
      If[var1Stop2Never3Use==9, Print[ebene[λ99,μ99]],
        leer=leer];
      gerade[t_]:=qs1+t(qs2-qs1);
      (* *) If[var1Stop2Never3Use==9, Print[gerade[t99]],leer=leer];
      sys[t_,λ_,μ_]:=(Table[
            ebene[λ,μ][[k]]==gerade[t][[k]],{k,1,Length[ps1]}]);
      (* *)  
      If[var1Stop2Never3Use==9, Print[sys[t99,λ99,μ99]],
        leer=leer];
      solv=Solve[sys[t,λ,μ],{t,λ,μ}]//Flatten;
      (* *)  If[var1Stop2Never3Use==9, Print[solv],leer=leer];
      solv1=solv[[1]][[2]];
      (* *)  If[var1Stop2Never3Use==9, Print[solv1],leer=leer];
      res=gerade[solv1]
      ];
    sch=  schnitt[pU1,pU2,pU3,pU7,pU8]