cp's OEIS Frontend

The triples (a, b, c) are displayed in increasing order of largest side c, and if largest sides c coincide then by increasing order of the middle side b.

Primitive triples means here that gcd(a, b, c, d) = 1 (see first example).

Equilateral triangles are not present because in this case O = I and d = 0.

Euler's triangle formula says that distance between the circumcenter O and the incenter I of a triangle is given by d = OI = sqrt(R*(R-2r)).

Heron's formula says the area A of a triangle whose sides have lengths a, b and c is given by A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2; then, the circumradius is given by R = abc/4A and the inradius r is given by r = A/s.

With these relations, d = OI = abc * sqrt(1/(16*A^2) - 1/(abc*(a+b+c))).

+-----+-----+-----+---------------+---------------+-----+----------------------+

| a | b | c | r | R | d | a+b+c| A |

+-----------+-----+---------------+---------------+-----+------+---------------+

| 10 | 10 | 16 | 8/3 | 25/3 | 5 | 36 | 48 |

| 40 | 40 | 48 | 12 | 25 | 5 | 128 | 768 |

| 16 | 49 | 55 | 11*sqrt(3)/3 | 49*sqrt(3)/3 | 21 | 120 | 220*sqrt(3) |

| 80 | 104 | 104 | 80/3 | 169/3 | 13 | 288 | 3840 |

| 15 | 169 | 176 | 11*sqrt(3)/3 | 169*sqrt(3)/3 | 91 | 360 | 1903sqrt(3)/3 |

| 130 | 130 | 240 | 24 | 169 | 143 | 500 | 6000 |

| 231 | 361 | 416 | 143*sqrt(3)/3 | 361*sqrt(3)/3 | 95 | 1008 | 24024*sqrt(3) |

| 246 | 246 | 480 | 80/3 | 1681/3 | 533 | 972 | 12960 |

| 272 | 272 | 480 | 60 | 289 | 221 | 1024 | 30720 |

| 480 | 510 | 510 | 144 | 289 | 17 | 1500 | 108000 |

| 296 | 296 | 560 | 140/3 | 1369/3 | 407 | 1152 | 26880 |

................................................................................

Observations coming from the previous table:

There exist two families of triangles,

1) triangle ABC is isosceles with a = b < c or a < b = c.

In this case, r and R are rational integers with same denominator = 1 or 3, and the area A of this triangle is a term of A231174.

Note that besides, if d is prime, d divides the two equal sides of the isosceles triangle, and also, there are these two possibilities:

-> d^2 = R and then r = (R-1)/2, or

-> d^2 = 3R and then r = (R-3)/2.

2) triangle ABC is scalene with a < b < c.

In this case, r and R are both quadratic of the form k*sqrt(3)/3.

For the corresponding primitive triples, miscellaneous properties and links, see A352314.

The sequence is not increasing. For example, a(2) = 128 for triangle with largest side = 48 while a(3) = 120 for triangle with largest side = 55.