solutions to  2 a^2 = b^2 + c^2
{{5, 1, 7}, {13, 7, 17}, {17, 7, 23}, {25, 17, 31}, {29, 1, 41}, {37, 23, 
    47}, {41, 31, 49}, {53, 17, 73}, {61, 49, 71}, {65, 23, 89}, {65, 47, 
    79}, {73, 7, 103}, {85, 41, 113}, {85, 71, 97}, {89, 41, 119}, {97, 7, 
    137}, {101, 79, 119}, {109, 31, 151}, {113, 97, 127}, {125, 73, 
    161}, {137, 17, 193}, {145, 119, 167}, {145, 127, 161}, {149, 89, 
    191}, {157, 47, 217}, {169, 1, 239}, {173, 113, 217}, {181, 161, 
    199}, {185, 49, 257}, {185, 119, 233}, {193, 73, 263}, {197, 167, 
    223}, {205, 23, 289}, {205, 103, 271}, {221, 31, 311}, {221, 199, 
    241}, {229, 161, 281}, {233, 103, 313}, {241, 89, 329}, {257, 223, 
    287}, {265, 151, 343}, {265, 241, 287}, {269, 191, 329}, {277, 137, 
    367}, {281, 71, 391}, {289, 79, 401}, {293, 217, 353}, {305, 17, 
    431}, {305, 137, 409}, {313, 287, 337}, {317, 233, 383}, {325, 49, 
    457}, {325, 287, 359}, {337, 113, 463}, {349, 119, 479}, {353, 47, 
    497}, {365, 281, 433}, {365, 337, 391}, {373, 23, 527}, {377, 193, 
    497}, {377, 217, 487}, {389, 151, 529}, {397, 97, 553}, {401, 359, 
    439}, {409, 271, 511}, {421, 391, 449}, {425, 7, 601}, {425, 329, 
    503}, {433, 263, 553}, {445, 193, 599}, {445, 353, 521}, {449, 71, 
    631}, {457, 257, 593}, {461, 119, 641}, {481, 41, 679}, {481, 449, 
    511}, {485, 383, 569}, {485, 439, 527}, {493, 313, 623}, {493, 343, 
    607}, {505, 41, 713}, {505, 239, 673}, {509, 239, 679}, {521, 161, 
    719}, {533, 127, 743}, {533, 433, 617}, {541, 79, 761}, {545, 329, 
    697}, {545, 511, 577}, {557, 367, 697}, {565, 7, 799}, {565, 217, 
    769}, {569, 289, 751}, {577, 527, 623}, {593, 97, 833}, {601, 311, 
    791}, {613, 577, 647}, {617, 503, 713}, {625, 191, 863}, {629, 31, 
    889}, {629, 521, 721}, {641, 409, 809}, {653, 257, 887}, {661, 289, 
    889}, {673, 167, 937}, {677, 623, 727}, {685, 511, 823}, {685, 647, 
    721}, {689, 161, 961}, {689, 569, 791}, {697, 73, 983}, {697, 487, 
    857}, {701, 391, 911}, {709, 401, 919}, {725, 263, 991}, {725, 311, 
    977}, {733, 617, 833}, {745, 497, 929}, {761, 721, 799}, {773, 553, 
    943}, {785, 727, 839}, {793, 607, 943}, {841, 799, 881}, {845, 713, 
    959}, {845, 721, 953}, {901, 839, 959}, {925, 881, 967}}

x = m^2 + 2mn - n^2
y = m^2 - 2mn - n^2
z = m^2 + n^2

cent[a_, b_, c_, d_] := (a^2 + b^2) (c^2 + d^2)
x1[a_, b_, c_, d_] := 
  Abs[(a c + b d)^2 + 2 (a c + b d) (a d - b c) - (a d - b c)^2 ]
y1[a_, b_, c_, d_] := 
  Abs[(a c + b d)^2 - 2 (a c + b d) (a d - b c) - (a d - b c)^2 ]
x2[a_, b_, c_, d_] := 
  Abs[ (a d + b c)^2 + 2 (a d + b c)  (a c - b d) - (a c - b d)^2]
y2[a_, b_, c_, d_] := 
  Abs[ (a d + b c)^2 - 2 (a d + b c)  (a c - b d) - (a c - b d)^2]
x3[a_, b_, c_, d_] := Abs[ (a^2 + b^2) (c^2 + 2c d - d^2)]
y3[a_, b_, c_, d_] := Abs[ (a^2 + b^2) (c^2 - 2c d - d^2)]
x4[a_, b_, c_, d_] := Abs[ (c^2 + d^2) (a^2 + 2a b - b^2)]
y4[a_, b_, c_, d_] := Abs[ (c^2 + d^2) (a^2 - 2a b - b^2)]
fullset[a_, b_, c_, d_] := {x1[a, b, c, d], x2[a, b, c, d], x3[a, b, c, d], 
    x4[a, b, c, d], cent[a, b, c, d], y4[a, b, c, d], y3[a, b, c, d], 
    y2[a, b, c, d], y1[a, b, c, d]}

fullset[1, 4, 3, 2]
{199, 31, 289, 91, 221, 299, 119, 311, 241}