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%I #41 Feb 16 2025 08:34:03
%S 10,10,16,40,40,48,16,49,55,80,104,104,15,169,176,130,130,240,231,361,
%T 416,246,246,480,272,272,480,480,510,510,296,296,560,350,350,672,455,
%U 961,1104,672,1200,1200,259,1040,1221,1040,1369,1551,1160,1160,1680,1218,1218,1680
%N 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 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 Primitive triples means here that gcd(a, b, c, d) = 1 (see first example).
%C Equilateral triangles are not present because in this case O = I and d = 0.
%C 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 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 With these relations, d = OI = abc * sqrt(1/(16*A^2) - 1/(abc*(a+b+c))).
%C +-----+-----+-----+---------------+---------------+-----+----------------------+
%C | a | b | c | r | R | d | a+b+c| A |
%C +-----------+-----+---------------+---------------+-----+------+---------------+
%C | 10 | 10 | 16 | 8/3 | 25/3 | 5 | 36 | 48 |
%C | 40 | 40 | 48 | 12 | 25 | 5 | 128 | 768 |
%C | 16 | 49 | 55 | 11*sqrt(3)/3 | 49*sqrt(3)/3 | 21 | 120 | 220*sqrt(3) |
%C | 80 | 104 | 104 | 80/3 | 169/3 | 13 | 288 | 3840 |
%C | 15 | 169 | 176 | 11*sqrt(3)/3 | 169*sqrt(3)/3 | 91 | 360 | 1903sqrt(3)/3 |
%C | 130 | 130 | 240 | 24 | 169 | 143 | 500 | 6000 |
%C | 231 | 361 | 416 | 143*sqrt(3)/3 | 361*sqrt(3)/3 | 95 | 1008 | 24024*sqrt(3) |
%C | 246 | 246 | 480 | 80/3 | 1681/3 | 533 | 972 | 12960 |
%C | 272 | 272 | 480 | 60 | 289 | 221 | 1024 | 30720 |
%C | 480 | 510 | 510 | 144 | 289 | 17 | 1500 | 108000 |
%C | 296 | 296 | 560 | 140/3 | 1369/3 | 407 | 1152 | 26880 |
%C ................................................................................
%C Observations coming from the previous table:
%C There exist two families of triangles,
%C 1) triangle ABC is isosceles with a = b < c or a < b = c.
%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.
%C 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 -> d^2 = R and then r = (R-1)/2, or
%C -> d^2 = 3R and then r = (R-3)/2.
%C 2) triangle ABC is scalene with a < b < c.
%C In this case, r and R are both quadratic of the form k*sqrt(3)/3.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Circumcircle.html">Circumcircle</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Circumradius.html">Circumradius</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerTriangleFormula.html">Euler Triangle Formula</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Inradius.html">Inradius</a>.
%e The table begins:
%e 10, 10, 16;
%e 40, 40, 48;
%e 16, 49, 55;
%e 80, 104, 104;
%e 15, 169, 176;
%e 130, 130, 240;
%e 231, 361, 416;
%e .........
%e 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 Cf. A231174, A350378, A350379, A352315.
%K nonn,tabf
%O 1,1
%A _Bernard Schott_, Mar 11 2022
%E More terms from _Jinyuan Wang_, Mar 12 2022