三顶点与等角点距离均有理的整边三角形

青青子衿

\begin{align*}\color{black}{
\left\{
\begin{split}
\large{x^2+xy+y^2}&\large{=a^2}\\
\large{y^2+yz+z^2}&\large{=b^2}\\
\large{x^2+xz+z^2}&\large{=c^2}
\end{split}\right.\qquad\,\begin{pmatrix}
\quad\,xyz\ne0\,\,\,;&\,\,\gcd(a,b,c)=1\,;&\qquad\,z>x>y\quad\qquad\,;\,\,\\
\,\,\,x,y,z\inQ\,;&\quad\,a,b,c\inN_+\quad;&\,0<a<b<c<1000\,.\,\,\\
\end{pmatrix}}
\end{align*}

\[ \color{black}{\begin{array}{cccc|cccc}  
\hline  
\phantom{*}\,\begin{matrix}      
\phantom{0}a\phantom{0}&\phantom{0}b\phantom{0}&\phantom{0}c\phantom{0}\\         
\end{matrix}&      
\phantom{+}x&\phantom{+}y&\phantom{+}z  
&\phantom{*}\,\begin{matrix}      
\phantom{0}a\phantom{0}&\phantom{0}b\phantom{0}&\phantom{0}c\phantom{0}\\         
\end{matrix}&      
\phantom{+}x&\phantom{+}y&\phantom{+}z  
\\   
\hline     
\phantom{*}\,\begin{matrix}      
\phantom{0}57&\phantom{0}65&\phantom{0}73\\         
\end{matrix}&      
\phantom{+}\tfrac{264}{7}&\phantom{+}\tfrac{195}{7}&\phantom{+}\tfrac{325}{7}
&\phantom{*}\,\begin{matrix}      
255&343&473\\         
\end{matrix}&  
\phantom{+}\tfrac{8415}{37}&\phantom{+}\tfrac{1785}{37}&\phantom{+}\tfrac{11704}{37}
\\   
\phantom{*}\,\begin{matrix}      
\phantom{0}73&\phantom{0}88&\phantom{0}95\\         
\end{matrix}&      
\phantom{+}\tfrac{325}{7}&\phantom{+}\tfrac{264}{7}   
&\phantom{+}\tfrac{440}{7}&\phantom{*}\,\begin{matrix}      
247&408&485\\         
\end{matrix}&     
\phantom{+}\tfrac{17575}{91}&\phantom{+}\tfrac{7752}{91}&\phantom{+}\tfrac{32640}{91}\\   
\phantom{*}\,\begin{matrix}      
\phantom{0}43&147&152\\         
\end{matrix}&     
\phantom{+}\tfrac{1064}{37}&\phantom{+}\tfrac{765}{37}&\phantom{+}\tfrac{5016}{37}   
&*\,\begin{matrix}      
152&365&497\\         
\end{matrix}&  
\phantom{+}\tfrac{28968}{169}&\small{-}\tfrac{8960}{169}&\phantom{+}\tfrac{65675}{169}\\  
\phantom{*}\,\begin{matrix}      
127&168&205\\         
\end{matrix}&      
\phantom{+}\tfrac{27265}{283}&\phantom{+}\tfrac{13464}{283}&\phantom{+}\tfrac{39360}{283}   
&*\,\begin{matrix}      
152&365&507\\         
\end{matrix}&      
\phantom{+}\tfrac{1224}{7}&\small{-}\tfrac{520}{7}&\phantom{+}\tfrac{2775}{7}\\   
\phantom{*}\,\begin{matrix}      
\phantom{0}97&185&208\\         
\end{matrix}&      
\phantom{+}\tfrac{6528}{91}&\phantom{+}\tfrac{3515}{91}&\phantom{+}\tfrac{14800}{91}   
&*\,\begin{matrix}      
217&425&608\\         
\end{matrix}&     
\phantom{+}\tfrac{50688}{211}&\small{-}\tfrac{12325}{211}&\phantom{+}\tfrac{95200}{211}\\   
\phantom{*}\,\begin{matrix}      
111&221&280\\         
\end{matrix}&   
\phantom{+}\tfrac{34200}{331}&\phantom{+}\tfrac{4641}{331}   
&\phantom{+}\tfrac{70720}{331}&\phantom{*}\,\begin{matrix}      
469&589&624\\         
\end{matrix}&      
\phantom{+}\tfrac{56448}{193}&\phantom{+}\tfrac{47957}{193}&\phantom{+}\tfrac{81840}{193}\\  
*\begin{matrix}      
\phantom{0}43&248&285\\         
\end{matrix}&     
\phantom{+}\tfrac{345}{7}&\small{-}\tfrac{136}{7}&\phantom{+}\tfrac{1800}{7}  
&*\,\begin{matrix}      
217&425&633\\         
\end{matrix}&      
\phantom{+}\tfrac{4752}{19}&\small{-}\tfrac{2125}{19}&\phantom{+}\tfrac{8925}{19}\\   
*\begin{matrix}      
\phantom{0}43&248&287\\         
\end{matrix}&   
\phantom{+}\tfrac{943}{19}&\small{-}\tfrac{448}{19}&\phantom{+}\tfrac{4920}{19}  
&*\,\begin{matrix}      
323&392&645\\         
\end{matrix}&  
\phantom{+}\tfrac{10455}{31}&\small{-}\tfrac{952}{31}&\phantom{+}\tfrac{12600}{31}\\   
\phantom{*}\,\begin{matrix}      
\phantom{0}49&285&296\\         
\end{matrix}&   
\phantom{+}\tfrac{12376}{331}&\phantom{+}\tfrac{5985}{331}&\phantom{+}\tfrac{91200}{331}  
&*\,\begin{matrix}      
323&392&713\\         
\end{matrix}&  
\phantom{+}\tfrac{16031}{43}&\small{-}\tfrac{7616}{43}&\phantom{+}\tfrac{19320}{43}\\  
\phantom{*}\,\begin{matrix}      
\phantom{0}95&312&343\\         
\end{matrix}&   
\phantom{+}\tfrac{1015}{13}&\phantom{+}\tfrac{360}{13}&\phantom{+}\tfrac{3864}{13}  
&*\,\begin{matrix}      
\phantom{0}57&673&715\\         
\end{matrix}&   
\phantom{+}\tfrac{825}{13}&\small{-}\tfrac{216}{13}&\phantom{+}\tfrac{8855}{13}\\   
\phantom{*}\,\begin{matrix}      
296&315&361\\         
\end{matrix}&     
\phantom{+}\tfrac{8512}{43}&\phantom{+}\tfrac{6120}{43}&\phantom{+}\tfrac{9405}{43}  
&*\,\begin{matrix}      
\phantom{0}57&673&728\\         
\end{matrix}&   
\phantom{+}\tfrac{840}{13}&\small{-}\tfrac{561}{13}&\phantom{+}\tfrac{9016}{13}\\   
\phantom{*}\,\begin{matrix}      
152&343&387\\         
\end{matrix}&  
\phantom{+}\tfrac{11520}{97}&\phantom{+}\tfrac{5096}{97}   
&\phantom{+}\tfrac{30429}{97}&\phantom{*}\,\begin{matrix}      
403&725&728\\         
\end{matrix}&  
\phantom{+}\tfrac{81928}{349}&\phantom{+}\tfrac{80475}{349}&\phantom{+}\tfrac{203000}{349}\\   
\phantom{*}\,\begin{matrix}      
323&392&407\\         
\end{matrix}&     
\phantom{+}\tfrac{8415}{43}&\phantom{+}\tfrac{7616}{43}&\phantom{+}\tfrac{11704}{43}   
&*\,\begin{matrix}      
245&632&817\\         
\end{matrix}&   
\phantom{+}\tfrac{25585}{97}&\small{-}\tfrac{4200}{97}&\phantom{+}\tfrac{63296}{97}\\   
\phantom{*}\,\begin{matrix}      
147&377&437\\         
\end{matrix}&      
\phantom{+}\tfrac{1656}{13}&\phantom{+}\tfrac{435}{13}&\phantom{+}\tfrac{4669}{13}   
&*\,\begin{matrix}      
245&632&873\\         
\end{matrix}&      
\phantom{+}\tfrac{5355}{19}&\small{-}\tfrac{3080}{19}&\phantom{+}\tfrac{13248}{19}\\   
\phantom{*}\,\begin{matrix}      
285&464&469\\         
\end{matrix}&  
\phantom{+}\tfrac{116025}{691}&\phantom{+}\tfrac{111360}{691}&\phantom{+}\tfrac{250096}{691}   
&\phantom{*}\,\begin{matrix}      
871&901&931\\         
\end{matrix}&   
\phantom{+}\tfrac{6765}{13}&\phantom{+}\tfrac{6307}{13}&\phantom{+}\tfrac{7208}{13}\\     
\hline     
\end{array}} \]

*  Note that the minus sign here indicates the distance on the extension line.
※※※※※※※※※※

1. AbsoluteTiming[nn = 300; lst = {};
2. Do[s = (a + b + c)/2;
3.   If[IntegerQ[2 s], area2 = s (s - a) (s - b) (s - c);
4.    m1 = (Sqrt[3] (b^2 + c^2 - a^2) + 4 Sqrt[area2])^2/(
5.     6 (a^2 + b^2 + c^2 + 4 Sqrt[3 area2]));
6.    m2 = (Sqrt[3] (a^2 + c^2 - b^2) + 4 Sqrt[area2])^2/(
7.     6 (a^2 + b^2 + c^2 + 4 Sqrt[3 area2]));
8.    m3 = (Sqrt[3] (a^2 + b^2 - c^2) + 4 Sqrt[area2])^2/(
9.     6 (a^2 + b^2 + c^2 + 4 Sqrt[3 area2]));
10.    If[0 < area2 && Sqrt[m1] != 0 && Sqrt[m2] != 0 && Sqrt[m3] != 0
11.        && (Element[Sqrt[m1], Rationals] &&  Element[Sqrt[m2], Rationals]
12.       && Element[Sqrt[m3], Rationals]) && GCD[a, b, c] == 1,
13.       AppendTo[lst, {a, b, c}]]], {a, nn}, {b, a}, {c, b}];
14.       Union[lst]]

复制代码

...
{Sqrt[#1^2 + #1*#2 + #2^2], Sqrt[#2^2 + #2*#3 + #3^2],
   Sqrt[#1^2 + #1*#3 + #3^2]} & @@@ {{264/7, 195/7, 325/7}, {325/7,
   264/7, 440/7}, {1064/37, 765/37, 5016/37}, {27265/283, 13464/283,
   39360/283}, {6528/91, 3515/91, 14800/91}, {34200/331, 4641/331,
   70720/331}, {345/7, -136/7, 1800/7}, {943/19, -448/19,
   4920/19}, {12376/331, 5985/331, 91200/331}, {1015/13, 360/13,
   3864/13}, {8512/43, 6120/43, 9405/43}, {11520/97, 5096/97,
   30429/97}, {8415/43, 7616/43, 11704/43}, {1656/13, 435/13,
   4669/13}, {116025/691, 111360/691, 250096/691}, {8415/37, 1785/37,
   11704/37}, {17575/91, 7752/91, 32640/91}, {28968/169, -8960/169,
   65675/169}, {1224/7, -520/7, 2775/7}, {50688/211, -12325/211,
   95200/211}, {56448/193, 47957/193, 81840/193}, {4752/19, -2125/19,
   8925/19}, {10455/31, -952/31, 12600/31}, {16031/43, -7616/43,
   19320/43}, {825/13, -216/13, 8855/13}, {840/13, -561/13,
   9016/13}, {81928/349, 80475/349, 203000/349}, {25585/97, -4200/97,
   63296/97}, {5355/19, -3080/19, 13248/19}, {6765/13, 6307/13,
   7208/13}}