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%I A352314 #40 Mar 15 2022 05:15:11
%S A352314 10,10,16,40,40,48,16,49,55,80,104,104,15,169,176,130,130,240,231,361,
%T A352314 416,246,246,480,272,272,480,480,510,510,296,296,560,350,350,672,455,
%U A352314 961,1104,672,1200,1200,259,1040,1221,1040,1369,1551,1160,1160,1680,1218,1218,1680
%N 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.
%C A352314 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.
%C A352314 Primitive triples means here that gcd(a, b, c, d) = 1 (see first example).
%C A352314 Equilateral triangles are not present because in this case O = I and d = 0.
%C A352314 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)).
%C A352314 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.
%C A352314 With these relations, d = OI = abc * sqrt(1/(16*A^2) - 1/(abc*(a+b+c))).
%C A352314 +-----+-----+-----+---------------+---------------+-----+----------------------+
%C A352314 |  a  |  b  |  c  |       r       |       R       |  d  | a+b+c|       A       |
%C A352314 +-----------+-----+---------------+---------------+-----+------+---------------+
%C A352314 |  10 |  10 |  16 |      8/3      |     25/3      |   5 |   36 |       48      |
%C A352314 |  40 |  40 |  48 |      12       |      25       |   5 |  128 |      768      |
%C A352314 |  16 |  49 |  55 |  11*sqrt(3)/3 | 49*sqrt(3)/3  |  21 |  120 |  220*sqrt(3)  |
%C A352314 |  80 | 104 | 104 |     80/3      |     169/3     |  13 |  288 |     3840      |
%C A352314 |  15 | 169 | 176 |  11*sqrt(3)/3 | 169*sqrt(3)/3 |  91 |  360 | 1903sqrt(3)/3 |
%C A352314 | 130 | 130 | 240 |      24       |      169      | 143 |  500 |     6000      |
%C A352314 | 231 | 361 | 416 | 143*sqrt(3)/3 | 361*sqrt(3)/3 |  95 | 1008 | 24024*sqrt(3) |
%C A352314 | 246 | 246 | 480 |     80/3      |     1681/3    | 533 |  972 |    12960      |
%C A352314 | 272 | 272 | 480 |      60       |      289      | 221 | 1024 |    30720      |
%C A352314 | 480 | 510 | 510 |     144       |      289      |  17 | 1500 |   108000      |
%C A352314 | 296 | 296 | 560 |    140/3      |     1369/3    | 407 | 1152 |    26880      |
%C A352314 ................................................................................
%C A352314 Observations coming from the previous table:
%C A352314 There exist two families of triangles,
%C A352314   1) triangle ABC is isosceles with a = b < c or a < b = c.
%C A352314 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.
%C A352314 Note that besides, if d is prime, d divides the two equal sides of the isosceles triangle, and also, there are these two possibilities:
%C A352314 -> d^2 = R and then r = (R-1)/2, or
%C A352314 -> d^2 = 3R and then r = (R-3)/2.
%C A352314   2) triangle ABC is scalene with a < b < c.
%C A352314 In this case, r and R are both quadratic of the form k*sqrt(3)/3.
%H A352314 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Circumcircle.html">Circumcircle</a>.
%H A352314 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Circumradius.html">Circumradius</a>.
%H A352314 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerTriangleFormula.html">Euler Triangle Formula</a>.
%H A352314 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%H A352314 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Inradius.html">Inradius</a>.
%e A352314 The table begins:
%e A352314    10,  10,  16;
%e A352314    40,  40,  48;
%e A352314    16,  49,  55;
%e A352314    80, 104, 104;
%e A352314    15, 169, 176;
%e A352314   130, 130, 240;
%e A352314   231, 361, 416;
%e A352314    .........
%e A352314 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.
%Y A352314 Cf. A231174, A350378, A350379, A352315.
%K A352314 nonn,tabf
%O A352314 1,1
%A A352314 _Bernard Schott_, Mar 11 2022
%E A352314 More terms from _Jinyuan Wang_, Mar 12 2022

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