Sequence of Triplets of Primitive Pythagorean Triangles with Same Area

Sequence
The sequence of sets of $3$ primitive Pythagorean triangles which all have the same area begins:


 * {| border="1"

! style = "padding: 2px 10px" | Generator ! style = "padding: 2px 10px" | Lengths of sides ! style = "padding: 2px 10px" | Area
 * align="right" style = "padding: 2px 10px" | $\left({77, 38}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({4485, 5852, 7373}\right)$
 * align="right" style = "padding: 2px 10px" | $13 \, 123 \, 110$
 * align="right" style = "padding: 2px 10px" | $\left({78, 55}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({3059, 8580, 9109}\right)$
 * align="right" style = "padding: 2px 10px" | $13 \, 123 \, 110$
 * align="right" style = "padding: 2px 10px" | $\left({138, 5}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({1380, 19 \, 019, 19 \, 069}\right)$
 * align="right" style = "padding: 2px 10px" | $13 \, 123 \, 110$
 * align="right" style = "padding: 2px 10px" | $\left({2035, 266}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({4 \, 070 \, 469, 1 \, 082 \, 620, 4 \, 163 \, 285}\right)$
 * align="right" style = "padding: 2px 10px" | $2 \, 203 \, 385 \, 574 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({3306, 61}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({10 \, 925 \, 915, 403 \, 332, 7 \, 309 \, 493}\right)$
 * align="right" style = "padding: 2px 10px" | $2 \, 203 \, 385 \, 574 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({3422, 55}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({11 \, 707 \, 059, 376 \, 420, 11 \, 713, 109}\right)$
 * align="right" style = "padding: 2px 10px" | $2 \, 203 \, 385 \, 574 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({1610, 869}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({1 \, 836 \, 939, 2 \, 798 \, 180, 3 \, 347 \, 261}\right)$
 * align="right" style = "padding: 2px 10px" | $2 \, 570 \, 042 \, 985 \, 510$
 * align="right" style = "padding: 2px 10px" | $\left({2002, 1817}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({706 \, 515, 7 \, 275 \, 268, 7 \, 309 \, 493}\right)$
 * align="right" style = "padding: 2px 10px" | $2 \, 570 \, 042 \, 985 \, 510$
 * align="right" style = "padding: 2px 10px" | $\left({2622, 143}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({6 \, 854 \, 435, 749 \, 892, 6 \, 895 \, 333}\right)$
 * align="right" style = "padding: 2px 10px" | $2 \, 570 \, 042 \, 985 \, 510$
 * align="right" style = "padding: 2px 10px" | $\left({2201, 1166}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({3 \, 484 \, 845, 5 \, 132 \, 732, 6 \, 203 \, 957}\right)$
 * align="right" style = "padding: 2px 10px" | $8 \, 943 \, 387 \, 723 \, 270$
 * align="right" style = "padding: 2px 10px" | $\left({2438, 2035}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({1 \, 802 \, 619, 9 \, 922 \, 660, 10 \, 085 \, 069}\right)$
 * align="right" style = "padding: 2px 10px" | $8 \, 943 \, 387 \, 723 \, 270$
 * align="right" style = "padding: 2px 10px" | $\left({3565, 198}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({12 \, 670 \, 021, 1 \, 411 \, 740, 12 \, 748 \, 429}\right)$
 * align="right" style = "padding: 2px 10px" | $8 \, 943 \, 387 \, 723 \, 270$
 * align="right" style = "padding: 2px 10px" | $\left({7238, 2465}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({46 \, 312 \, 419, 35 \, 683 \, 340, 6 \, 203 \, 957}\right)$
 * align="right" style = "padding: 2px 10px" | $826 \, 290 \, 896 \, 699 \, 730$
 * align="right" style = "padding: 2px 10px" | $\left({9077, 1122}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({81 \, 133 \, 045, 20 \, 368 \, 788, 10 \, 085 \, 069}\right)$
 * align="right" style = "padding: 2px 10px" | $826 \, 290 \, 896 \, 699 \, 730$
 * align="right" style = "padding: 2px 10px" | $\left({10 \, 434, 731}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({108 \, 333 \, 995, 15 \, 254 \, 508, 12 \, 748 \, 429}\right)$
 * align="right" style = "padding: 2px 10px" | $826 \, 290 \, 896 \, 699 \, 730$
 * align="right" style = "padding: 2px 10px" | $\left({352 \, 538, 2 \, 999 \, 447}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({8 \, 872 \, 399 \, 264 \, 365, 2 \, 114 \, 838 \, 092 \, 972, 9 \, 120 \, 965 \, 347 \, 253}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({1 \, 931 \, 103, 2 \, 398 \, 838}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({2 \, 025 \, 264 \, 953 \, 635, 9 \, 264 \, 806 \, 516 \, 628, 9 \, 483 \, 582 \, 546 \, 853}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({3 \, 063 \, 347, 3 \, 215 \, 070}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({952 \, 580 \, 262 \, 491, 19 \, 697 \, 750 \, 078 \, 580, 19 \, 720 \, 769 \, 947 \, 309}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * }
 * align="right" style = "padding: 2px 10px" | $826 \, 290 \, 896 \, 699 \, 730$
 * align="right" style = "padding: 2px 10px" | $\left({10 \, 434, 731}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({108 \, 333 \, 995, 15 \, 254 \, 508, 12 \, 748 \, 429}\right)$
 * align="right" style = "padding: 2px 10px" | $826 \, 290 \, 896 \, 699 \, 730$
 * align="right" style = "padding: 2px 10px" | $\left({352 \, 538, 2 \, 999 \, 447}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({8 \, 872 \, 399 \, 264 \, 365, 2 \, 114 \, 838 \, 092 \, 972, 9 \, 120 \, 965 \, 347 \, 253}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({1 \, 931 \, 103, 2 \, 398 \, 838}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({2 \, 025 \, 264 \, 953 \, 635, 9 \, 264 \, 806 \, 516 \, 628, 9 \, 483 \, 582 \, 546 \, 853}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * align="right" style = "padding: 2px 10px" | $\left({3 \, 063 \, 347, 3 \, 215 \, 070}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({952 \, 580 \, 262 \, 491, 19 \, 697 \, 750 \, 078 \, 580, 19 \, 720 \, 769 \, 947 \, 309}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * }
 * align="right" style = "padding: 2px 10px" | $\left({3 \, 063 \, 347, 3 \, 215 \, 070}\right)$
 * align="right" style = "padding: 2px 10px" | $\left({952 \, 580 \, 262 \, 491, 19 \, 697 \, 750 \, 078 \, 580, 19 \, 720 \, 769 \, 947 \, 309}\right)$
 * align="right" style = "padding: 2px 10px" | $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
 * }

These appear to be the only ones known.