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Mathematica

aMax=360(*WARNING:takes a long time*);
heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
cayley=1/24Sqrt[2Det[{
  {0,1,1,1,1},
  {1,0,#1^2,#2^2,#6^2},
  {1,#1^2,0,#3^2,#5^2},
  {1,#2^2,#3^2,0,#4^2},
  {1,#6^2,#5^2,#4^2,0}
}]]&;
Do[
  S1=heron[a,b,c];
  If[S1//IntegerQ//Not,Continue[]];
  Do[
    S2=heron[a,e,f];
    If[S2//IntegerQ//Not,Continue[]];
    Do[
      If[GCD[a, b, c, d, e, f] > 1, Continue[]];
      If[b==e&&c>f||b==f&&c>e,Continue[]];
      S3=heron[b,d,f];
      If[S3//IntegerQ//Not,Continue[]];
      S4=heron[c,d,e];
      If[S4//IntegerQ//Not,Continue[]];
      V=cayley[a,b,c,d,e,f];
      If[V//IntegerQ//Not,Continue[]];
      If[V==0,Continue[]];
      a//Sow(*{a,b,c,d,e,f,S1,S2,S3,S4,V}//Sow*);
    ,{d,Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1-S2)/a)^2]//Ceiling,Min[a,Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1+S2)/a)^2]]}];
  ,{e,a-b+1,b},{f,a-e+1,b}];
,{a,117,aMax},{b,a/2//Ceiling,a},{c,a-b+1,b}]//Reap//Last//Last