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I use proflycee to make a tex file like this

    \documentclass[12pt,a4paper]{article}
    \usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
    \usepackage{nicefrac}
    \usepackage[build]{ProfLycee}
    \usepackage{mathrsfs}
    \usepackage{amsmath}
    \usepackage{amsthm}
    \theoremstyle{definition}
    \newtheorem{ex}{Exercise}
    \begin{document}
        \begin{ex}
    Une équation cartésienne du plan  passant par le points 
    $A\AffPoint(3/2,-2/3,-4/5)$,
    $B\AffPoint(8,1,5)$ et $C\AffPoint(9,3,6)$ est
    \[ \mathcal{R} \text{ : } \TrouveEqCartPlan[SimplifCoeffs,Facteur=-1](3/2,-2/3,-4/5)(8,1,5)(9,3,6).\]
        \end{ex}
    
        \begin{ex}
    Une équation cartésienne du plan  passant par le points 
    $A\AffPoint(1,0,0)$,
    $B\AffPoint(0,1,0)$ et $C\AffPoint(0,0,1)$ est
    \[ \mathcal{R} \text{ : } \TrouveEqCartPlan[SimplifCoeffs](1,0,0)(0,1,0)(0,0,1).\]
\end{ex}
    \end{document}

I got

enter image description here

With Mathematica, I tried

SetDirectory[NotebookDirectory[]];
Clear["Global`*"];
pA = {3/2, -2/3, -4/5};
pB = {8, 1, 5};
pC = {9, 3, 6};
myeq = TraditionalForm[
    Subtract @@ Expand[CoplanarPoints[{pA, pB, pC, {x, y, z}}]]] == 
   0;
    toX[e_] := 
  StringReplace[ToString[TeXForm[e]], {"\\}" -> ")", "\\{" -> "("}];
rep[s_String] := StringReplace[s, "," -> ";"];

fileName = FileNameJoin[{Directory[], "three_points_24_06_2024.tex"}]
If[FileExistsQ[fileName], DeleteFile[fileName]];
file = OpenWrite[fileName, PageWidth -> Infinity];
WriteString[file, 
  "\\documentclass[12pt,a4paper]{article}\n" <> 
   "\\usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}\n\
" <> "\\usepackage{amsmath}\n" <> "\\usepackage{amsthm}\n" <> 
   "\\usepackage{esvect}\n" <> "\\theoremstyle{definition}\n" <> 
   "\\newtheorem{ex}{Exercise}\n" <> "\\begin{document}\n"];
WriteString[file, 
  "\\begin{ex}\n Une équation cartésienne du plan  passant par le \
points  $A" <> rep@toX[pA] <> "$ et $B" <> rep@toX[pB] <> "$, $C" <> 
   rep@toX[pC] <> "$ est $(P):" <> toX[myeq] <> ".$ \n\\end{ex}\n"
  ];
WriteString[file, "\\end{document}\n"];
Close[file]

The TeX file I got

    \documentclass[12pt,a4paper]{article}
    \usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
    \usepackage{amsmath}
    \usepackage{amsthm}
    \usepackage{esvect}
    \theoremstyle{definition}
    \newtheorem{ex}{Exercise}
    \begin{document}
    \begin{ex}
     Une e'quation carte'sienne du plan  passant par le points  $A\left(\frac{3}{2};-\frac{2}{3};-\frac{4}{5}\right)$ et $B(8;1;5)$, $C(9;3;6)$ est $(P):298 x+21 y-340 z-705=0.$ 
    \end{ex}
    \end{document}

and PDF file

enter image description here

With two Exercises, I tried

SetDirectory[NotebookDirectory[]];
Clear["Global`*"];
pA = {3/2, -2/3, -4/5};
pB = {8, 1, 5};
pC = {9, 3, 6};
myeq = TraditionalForm[
    Subtract @@ Expand[CoplanarPoints[{pA, pB, pC, {x, y, z}}]]] == 
   0;
pA1 = {1, 0, 0};
pB1 = {0, 1, 0};
pC1 = {0, 0, 1};
myeq1 = TraditionalForm[
    Subtract @@ 
     Expand[CoplanarPoints[{pA1, pB1, pC1, {x, y, z}}]]] == 0;
toX[e_] := 
  StringReplace[ToString[TeXForm[e]], {"\\}" -> ")", "\\{" -> "("}];
rep[s_String] := StringReplace[s, "," -> ";"];
fileName = FileNameJoin[{Directory[], "three_points_24_06_2024.tex"}]
If[FileExistsQ[fileName], DeleteFile[fileName]];
file = OpenWrite[fileName, PageWidth -> Infinity];
WriteString[file, 
  "\\documentclass[12pt,a4paper]{article}\n" <> 
   "\\usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}\n\
" <> "\\usepackage{amsmath}\n" <> "\\usepackage{amsthm}\n" <> 
   "\\usepackage{esvect}\n" <> "\\theoremstyle{definition}\n" <> 
   "\\newtheorem{ex}{Exercise}\n" <> "\\begin{document}\n"];
WriteString[file, 
  "\\begin{ex}\n Une équation cartésienne du plan  passant par le \
points  $A" <> rep@toX[pA] <> "$ et $B" <> rep@toX[pB] <> "$, $C" <> 
   rep@toX[pC] <> "$ est $(P):" <> toX[myeq] <> ".$ \n\\end{ex}\n
\\begin{ex}\n Une équation cartésienne du plan  passant par le points \
 $A" <> rep@toX[pA1] <> "$ et $B" <> rep@toX[pB1] <> "$, $C" <> 
   rep@toX[pC1] <> "$ est $(P):" <> toX[myeq1] <> ".$ \n\\end{ex}\n
"
  ];
WriteString[file, "\\end{document}\n"];
Close[file]

I got

    \documentclass[12pt,a4paper]{article}
    \usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
    \usepackage{amsmath}
    \usepackage{amsthm}
    \usepackage{esvect}
    \theoremstyle{definition}
    \newtheorem{ex}{Exercise}
    \begin{document}
    \begin{ex}
     Une e'quation carte'sienne du plan  passant par le points  $A\left(\frac{3}{2};-\frac{2}{3};-\frac{4}{5}\right)$ et $B(8;1;5)$, $C(9;3;6)$ est $(P):\text{myeq}.$ 
    \end{ex}
    
    \begin{ex}
 Une e'quation carte'sienne du plan  passant par le points  $A(1;0;0)$ et $B(0;1;0)$, $C(0;0;1)$ est $(P):x+y+z-1=0.$ 
\end{ex}
    
    \end{document}

enter image description here

How can I make a list of exercises from a list of triples?

list = {{{3/2, -2/3, -4/5}, {8, 1, 5}, {9, 3, 6}}, {{1, 0, 0}, {0, 1, 
    0}, {0, 0, 1}}};
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2 Answers 2

Reset to default
1
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It is almost here

SetDirectory[NotebookDirectory[]]
lis = {{{3/2, -2/3, -4/5}, {8, 1, 5}, {9, 3, 6}}, {{1, 0, 0}, {0, 1, 
     0}, {0, 0, 1}}};
data = Table[{pA, pB, pC} = points;
  mylist = {pA, pB, pC, 
    Subtract @@ Expand[CoplanarPoints[{pA, pB, pC, {x, y, z}}]] == 
     0}, {points, lis}]
toX[e_] := 
  StringReplace[
   ToString[TeXForm[e]], {"\\}" -> ")", "\\{" -> "(", 
    "\\frac" -> "\\dfrac"}];
rep[s_String] := StringReplace[s, "," -> ";"];
fileName = FileNameJoin[{Directory[], "three_points_24_06_2024.tex"}]
If[FileExistsQ[fileName], DeleteFile[fileName]];
file = OpenWrite[fileName, PageWidth -> Infinity];
WriteString[file, 
 "\\documentclass[12pt,a4paper]{article}\n" <> 
  "\\usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}\n\
" <> "\\usepackage{amsmath}\n" <> "\\usepackage{amsthm}\n" <> 
  "\\usepackage{esvect}\n" <> "\\theoremstyle{definition}\n" <> 
  "\\newtheorem{ex}{Exercise}\n" <> "\\begin{document}\n"]; Do[
 WriteString[file, 
  "\\begin{ex}\n" <> 
   "Une e'quation carte'sienne du plan  passant par le points $A" <> 
   rep@toX[data[[n, 1]]] <> "$, $B" <> rep@toX[data[[n, 2]]] <> 
   "$ et $C" <> rep@toX[data[[n, 3]]] <> "$ est $" <> 
   toX[data[[n, 4]]] <> ".$\n\\end{ex}\n"], {n, 1, Length@data}];
WriteString[file, "\\end{document}\n"];
Close[file]

The TeX file

    \documentclass[12pt,a4paper]{article}
    \usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
    \usepackage{amsmath}
    \usepackage{amsthm}
    \usepackage{esvect}
    \theoremstyle{definition}
    \newtheorem{ex}{Exercise}
    \begin{document}
    \begin{ex}
    Une e'quation carte'sienne du plan  passant par le points $A\left(\dfrac{3}{2};-\dfrac{2}{3};-\dfrac{4}{5}\right)$, $B(8;1;5)$ et $C(9;3;6)$ est $298 x+21 y-340 z-705=0.$
    \end{ex}
    \begin{ex}
    Une e'quation carte'sienne du plan  passant par le points $A(1;0;0)$, $B(0;1;0)$ et $C(0;0;1)$ est $x+y+z-1=0.$
    \end{ex}
    \end{document}

enter image description here

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Are you looking for a list of equations for the plane?

plane[pts_] := 
 Det[ Prepend[ 
   Append[#, 1] & /@ pts, {\[FormalX], \[FormalY], \[FormalZ], 1}]]

plane /@ list
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