A triangle with both natural number side lengths and natural number area is called Heronic after Heron, Hero of Alexandria. The area of Heronic triangle is always a multiple of six.
I made a list of Heronic Triangles up to sides 10,000 for another answer. From there, here are the Heronic triangles whose areas equal three times their perimeters. The list looks like it has a≤b≤c.a≤b≤c. Only one area appears twice, and that’s for non congruent triangles. so none of these are congruent. The answer is at least 45 such triangles. I’m thinking it’s infinite. They get pointier and pointier but I bet they just keep coming.
Here are the 45 on my list:
a b c area
20 20 24 192
17 25 26 204
17 25 28 210
20 21 29 210
18 24 30 216
16 30 34 240
15 34 35 252
15 36 39 270
22 26 40 264
14 48 50 336
27 30 51 324
25 33 52 330
24 35 53 336
21 45 60 378
16 52 60 384
20 51 65 408
14 61 65 420
19 60 73 456
38 40 74 456
35 44 75 462
32 50 78 480
30 56 82 504
29 60 85 522
13 84 85 546
18 75 87 540
28 65 89 546
26 80 102 624
25 92 113 690
17 105 116 714
13 122 125 780
24 110 130 792
14 130 136 840
73 74 145 876
23 140 159 966
55 111 164 990
49 148 195 1176
16 195 205 1248
22 200 218 1320
46 185 229 1380
43 259 300 1806
21 380 397 2394
41 370 409 2460
40 481 519 3120
39 703 740 4446
38 1369 1405 8436
