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How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to 3 times their perimeters?

May 1, 2020

#1
+79
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Not sure how to help mate but check out this website:

https://www.quora.com/How-many-non-congruent-triangles-with-positive-integer-leg-lengths-have-areas-that-are-numerically-equal-to-3-times-their-perimeter

May 1, 2020
#2
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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

May 1, 2020