Here’s a set of three triangles:

(24/5, 35/12, 337/60)

area = 7, perimeter = 40/3

(8/3, 21/2, 65/6)

area = 14, perimeter = 24

(12, 7/2, 25/2)

area = 21, perimeter = 28

whose sums by twos’ are the sides 21, 28, 35 of a right triangle

(21, 28, 35) = 7 × [3, 4, 5]

area = 294, perimeter = 84

Find more sets of 3 triangles the sum of whose areas by twos are sides of a right triangle.

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Here are a few more, Format is

{{a, b, c, area },{d, e, f, area},{g, h, I, area}}

{{6,8,10,24},{5,12,13,30},{18,24,30,216}}

24 + 30 = 54 , 24 + 216 = 240 , 30 + 216 = 246

Right Triangle (54, 240, 246) = 6 x (9, 40, 41)

{{8,15,17,60},{10,24,26,120},{9,40,41,180}}

60 + 120 = 180 , 60 + 180 = 240 , 120 + 180 = 300

Right Triangle (180, 240, 300) = 60 x (3, 4, 5)

{{12,16,20,96},{10,24,26,120},{36,48,60,864}}

96 + 120 = 216 , 96 + 864 = 960 , 120 + 864 = 984

Right Triangle (216, 960, 984) = 24 x (9, 40, 41)

{{10,24,26,120},{15,20,25,150},{30,72,78,1080}}

120 + 150 = 270 , 120 + 1080 = 1200 , 150 + 1080 = 1230

Right Triangle (270, 1200, 1230) = 30 x (9, 40, 41)

{{10,24,26,120},{9,40,41,180},{30,40,50,600}}

120 + 180 = 300 , 120 + 600 = 720 , 180 + 600 = 780

Right Triangle (300, 720, 780) = 60 x (5, 12, 13)

{{18,24,30,216},{16,63,65,504},{24,45,51,540}}

216 + 504 = 720 , 216 + 540 = 756 , 504 + 540 = 1044

Right Triangle (720, 756, 1044) = 36 x (20, 21, 29)

{{16,30,34,240},{15,36,39,270},{32,255,257,4080}}

240 + 270 = 510 , 240 + 4080 = 4320 , 270 + 4080 = 4350

Right Triangle (510, 4320, 4350) = 30 x (17, 144, 145)

{{16,30,34,240},{20,48,52,480},{18,80,82,720}}

240 + 480 = 720 , 240 + 720 = 960 , 480 + 720 = 1200

Right Triangle (720, 960, 1200) = 240 x (3, 4, 5)

{{14,48,50,336},{18,80,82,720},{33,56,65,924}}

336 + 720 = 1056 , 336 + 924 = 1260 , 720 + 924 = 1644

Right Triangle (1056, 1260, 1644) = 12 x (88, 105, 137)

{{16,63,65,504},{28,45,53,630},{48,189,195,4536}}

504 + 630 = 1134 , 504 + 4536 = 5040 , 630 + 4536 = 5166

Right Triangle (1134, 5040, 5166) = 126 x (9, 40, 41)

{{16,63,65,504},{15,112,113,840},{32,126,130,2016}}

504 + 840 = 1344 , 504 + 2016 = 2520 , 840 + 2016 = 2856

Right Triangle (1344, 2520, 2856) = 168 x (8, 15, 17)

{{16,63,65,504},{24,70,74,840},{32,126,130,2016}}

504 + 840 = 1344 , 504 + 2016 = 2520 , 840 + 2016 = 2856

Right Triangle (1344, 2520, 2856) = 168 x (8, 15, 17)

{{16,63,65,504},{40,42,58,840},{32,126,130,2016}}

504 + 840 = 1344 , 504 + 2016 = 2520 , 840 + 2016 = 2856

Right Triangle (1344, 2520, 2856) = 168 x (8, 15, 17)

{{16,63,65,504},{30,72,78,1080},{36,77,85,1386}}

504 + 1080 = 1584 , 504 + 1386 = 1890 , 1080 + 1386 = 2466

Right Triangle (1584, 1890, 2466) = 18 x (88, 105, 137)

{{24,45,51,540},{30,72,78,1080},{27,120,123,1620}}

540 + 1080 = 1620 , 540 + 1620 = 2160 , 1080 + 1620 = 2700

Right Triangle (1620, 2160, 2700) = 540 x (3, 4, 5)

{{13,84,85,546},{15,112,113,840},{44,117,125,2574}}

546 + 840 = 1386 , 546 + 2574 = 3120 , 840 + 2574 = 3414

Right Triangle (1386, 3120, 3414) = 6 x (231, 520, 569)

{{13,84,85,546},{24,70,74,840},{44,117,125,2574}}

546 + 840 = 1386 , 546 + 2574 = 3120 , 840 + 2574 = 3414

Right Triangle (1386, 3120, 3414) = 6 x (231, 520, 569)

{{13,84,85,546},{40,42,58,840},{44,117,125,2574}}

546 + 840 = 1386 , 546 + 2574 = 3120 , 840 + 2574 = 3414

Right Triangle (1386, 3120, 3414) = 6 x (231, 520, 569)

{{30,40,50,600},{30,72,78,1080},{35,120,125,2100}}

600 + 1080 = 1680 , 600 + 2100 = 2700 , 1080 + 2100 = 3180

Right Triangle (1680, 2700, 3180) = 60 x (28, 45, 53)

{{30,40,50,600},{45,60,75,1350},{39,80,89,1560}}

600 + 1350 = 1950 , 600 + 1560 = 2160 , 1350 + 1560 = 2910

Right Triangle (1950, 2160, 2910) = 30 x (65, 72, 97)

{{21,72,75,756},{33,56,65,924},{45,336,339,7560}}

756 + 924 = 1680 , 756 + 7560 = 8316 , 924 + 7560 = 8484

Right Triangle (1680, 8316, 8484) = 84 x (20, 99, 101)

{{15,112,113,840},{36,105,111,1890},{26,168,170,2184}}

840 + 1890 = 2730 , 840 + 2184 = 3024 , 1890 + 2184 = 4074

Right Triangle (2730, 3024, 4074) = 42 x (65, 72, 97)

{{24,70,74,840},{36,105,111,1890},{26,168,170,2184}}

840 + 1890 = 2730 , 840 + 2184 = 3024 , 1890 + 2184 = 4074

Right Triangle (2730, 3024, 4074) = 42 x (65, 72, 97)

{{40,42,58,840},{36,105,111,1890},{26,168,170,2184}}

840 + 1890 = 2730 , 840 + 2184 = 3024 , 1890 + 2184 = 4074

Right Triangle (2730, 3024, 4074) = 42 x (65, 72, 97)

{{36,48,60,864},{32,126,130,2016},{48,90,102,2160}}

864 + 2016 = 2880 , 864 + 2160 = 3024 , 2016 + 2160 = 4176

Right Triangle (2880, 3024, 4176) = 144 x (20, 21, 29)

{{32,60,68,960},{40,96,104,1920},{36,160,164,2880}}

960 + 1920 = 2880 , 960 + 2880 = 3840 , 1920 + 2880 = 4800

Right Triangle (2880, 3840, 4800) = 960 x (3, 4, 5)

{{20,99,101,990},{30,72,78,1080},{45,1012,1013,22770}}

990 + 1080 = 2070 , 990 + 22770 = 23760 , 1080 + 22770 = 23850

Right Triangle (2070, 23760, 23850) = 90 x (23, 264, 265)

{{30,72,78,1080},{36,105,111,1890},{40,198,202,3960}}

1080 + 1890 = 2970 , 1080 + 3960 = 5040 , 1890 + 3960 = 5850

Right Triangle (2970, 5040, 5850) = 90 x (33, 56, 65)

{{40,75,85,1500},{50,120,130,3000},{45,200,205,4500}}

1500 + 3000 = 4500 , 1500 + 4500 = 6000 , 3000 + 4500 = 7500

Right Triangle (4500, 6000, 7500) = 1500 x (3, 4, 5)

Paul.