A352314 Primitive triples (a, b, c) of integer-sided triangles such that the distance d = OI between the circumcenter O and the incenter I is also a positive integer. The triples of sides (a, b, c) are in increasing order a <= b <= c.

10, 10, 16, 40, 40, 48, 16, 49, 55, 80, 104, 104, 15, 169, 176, 130, 130, 240, 231, 361, 416, 246, 246, 480, 272, 272, 480, 480, 510, 510, 296, 296, 560, 350, 350, 672, 455, 961, 1104, 672, 1200, 1200, 259, 1040, 1221, 1040, 1369, 1551, 1160, 1160, 1680, 1218, 1218, 1680

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..54.

Eric Weisstein's World of Mathematics, Circumcircle.

Eric Weisstein's World of Mathematics, Circumradius.

Eric Weisstein's World of Mathematics, Euler Triangle Formula.

Eric Weisstein's World of Mathematics, Incircle.

Eric Weisstein's World of Mathematics, Inradius.

Example

The table begins:
   10,  10,  16;
   40,  40,  48;
   16,  49,  55;
   80, 104, 104;
   15, 169, 176;
  130, 130, 240;
  231, 361, 416;
   .........
For first triple (10, 10, 16), s = (10+10+16)/2 = 18, A = 48, r = 48/18 = 8/3, R = 10*10*16/4*48 = 25/3, and d = sqrt(25/3 * 9/3) = 5. We observe that gcd(10, 10, 16) = 2, but that gcd(10, 10, 16, 5) = 1, in fact for triple (5, 5, 8) with gcd(5, 5, 8) = 1, OI should be 5/2.

Crossrefs

Cf. A231174, A350378, A350379, A352315.

Sequence in context: A272479 A167427 A227437 * A363328 A047879 A344336

Adjacent sequences: A352311 A352312 A352313 * A352315 A352316 A352317

Keyword

nonn,tabf

Author

Bernard Schott, Mar 11 2022

Extensions

More terms from Jinyuan Wang, Mar 12 2022

Status

approved