# Properties of triangles with integer sides and area

OEIS sequence A051518 describes

There exists a triangle of perimeter $n$ having integer sides and area.

And begins

12, 16, 18, 24, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 96, 98, 100, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196, 198, 200, 204


Uncharacteristically, the OEIS sequence doesn't have any comments describing the sequence. I'd like to know more about this sequence and the underlying triangles. For example:

• Must all of the terms be even?

• Are there infinitely many even numbers that do not appear in the sequence?

• Are there infinitely many terms $a(n) = a(n - 1) + 6$?

• How many such triangles exist for a given $n$? (e.g. There are at least 4 non-congruent triangles with a perimeter of $a(6)=36$ and integer area.)

• What is the asymptotic growth of such multiplicity?

• What is the ratio of right vs isosceles vs scalene triangles with perimeter less than $N$?

• Can all triangles with integer perimeter and area be described with integer coordinates? (e.g. the $7$-$15$-$20$ triangle can be described with $(0, 0), (12, 16), (0, 7)$.)

Here's what I know (and I may have missed some cases)

a(1)  = 12 describes the 3-4-5    right     triangle.
a(2)  = 16 describes the 5-5-6    isosceles triangle.
a(3)  = 18 describes the 5-5-8    isosceles triangle.
a(4)  = 24 describes the 6-8-10   right     triangle (scaling of a(1)).
a(5)  = 30 describes the 5-12-13  tright    triangle.
a(6)  = 32 describes the 10-10-12 isosceles triangle (scaling of a(2)) or
the 4-13-15  scalene   triangle.

a(7)  = 36 describes the 10-13-13  isosceles triangle or
the 10-10-16 isosceles triangle (scaling of a(3)) or
the 9-12-15  isosceles triangle (scaling of a(1)) or
the 9-10-17  scalene   triangle.

a(8)  = 40 describes the 8-15-17  right     triangle.
a(9)  = 42 describes the 7-15-20  scalene   triangle or
the 13-14-15 scalene   triangle.

a(10) = 44 describes the 11-13-20 scalene   triangle.
a(11) = 48 describes the 15-15-18 isosceles triangle (scaling of a(2)) or
the 10-17-21 scalene   triangle or
the 12-16-20 right     triangle (scaling of a(1)).


Perhaps this is too open ended to be a good question for MSE.

• Heron's formula may offer some insight. For instance, the semiperimeter features prominently in some forms of the formula, that can probably be used to show that $n$ must be even. – Arthur Jan 15 '18 at 7:19
• Ah, indeed Heron's formula proves that the semiperimeter must be an integer—and so the perimeter must be even. – Peter Kagey Jan 15 '18 at 7:29
• You might want to see Heronian triangle. Also, see here. – mathlove Jan 15 '18 at 12:49

### Must all of the terms be even?

Yes, as Arthur suggested, Heron's formula states that if the sides of the triangles are $$a$$, $$b$$, and $$c$$, with semiperimeter $$s$$, then the area is $$A = \sqrt{s(s-a)(s-b)(s-c)}.$$ If the semiperimeter were not an integer, then $$A$$ would not be rational—thus the semiperimeter is an integer and the perimeter is even.

### Are there infinitely many even numbers that do not appear in the sequence?

Yes! For example, $$2p$$ does not appear in the sequence for any prime $$p$$. By Heron's formula, in order for the area $$A$$ to be an integer, $$s(s-a)(s-b)(s-c)$$ must be square. But this means (without loss of generality) that $$s-a = p = a$$. However, this violates the triangle inequality of Euclidean geometry.

In any case, I created OEIS sequence A305703, which lists the even numbers that do not appear in the sequence.

### Are there infinitely many terms $$a(n)=a(n−1)+6?$$

I don't know. But Zhang's proof of bounded gaps between primes implies that $$a(n)=a(n-1) + k$$ for some $$k$$.

### How many such triangles exist for a given n? (e.g. There are at least 4 non-congruent triangles with a perimeter of a(6)=36 and integer area.)

I don't know, but I've added this to the OEIS as A305717. It looks like this may not have a lot of structure.

### What is the asymptotic growth of such multiplicity?

This is a good question for the aforementioned OEIS sequence.

### What is the ratio of right vs isosceles vs scalene triangles with perimeter less than N?

Again, I don't know, but this could also spin-off into a few good OEIS sequences.

### Can all triangles with integer perimeter and area be described with integer coordinates? (e.g. the 7-15-20 triangle can be described with (0,0),(12,16),(0,7).)

Yes! Paul Yiu had a nice article in the March 2001 edition of The American Mathematical Monthly called Heronian Triangles Are Lattice Triangles with a neat proof.