A350378 -id:A350378 - cp's OEIS frontend

A350379 Consider integer-sided triangles having integer area and let d be the distance between the center of the inscribed circle and the center of the circumscribed circle. The sequence lists the set of distances d which are prime numbers.

5, 13, 17, 53, 193, 241

Offset: 1

Michel Lagneau, Dec 28 2021

The corresponding integer areas of integer-sided triangles such that the distance between the incenter and the circumcenter is a prime number is given by the sequence A350378.
In geometry, Euler's theorem states that the distance between the incenter and circumcenter can be expressed as d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius.
The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.
The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.
The following table gives the first values (A, a, b, c, r, R, d) where A is the area of the triangles, a, b, c are the integer sides of the triangles, r is the inradius, R is the circumradius and d is the distance between the incenter and circumcenter with d = sqrt(R(R-2r)).
+------------+--------+--------+---------+---------+---------+-----+
|         A  |     a  |     b  |     c   |    r    |    R    |  d  |
+------------+--------+--------+---------+---------+---------+-----+
|         48 |     10 |     10 |     16  |    8/3  |    25/3 |   5 |
|        768 |     40 |     40 |     48  |   12    |    25   |   5 |
|       3840 |     80 |    104 |    104  |   80/3  |   169/3 |  13 |
|     108000 |    480 |    510 |    510  |  144    |   289   |  17 |
|    1134000 |   1590 |   1590 |   1680  |  1400/3 |  2809/3 |  53 |
|  200202240 |  21280 |  21616 |  21616  | 18620/3 | 37249/3 | 193 |
| 4382077920 | 100320 | 100738 | 100738  | 29040   | 58081   | 241 |
....................................................................
From the previous table, we observe that the triangles are isosceles, the distance between the incenter and the circumcenter is d = sqrt(R) if R is a perfect square, or d = sqrt(3R) if R is of the form k^2/3, k integer. We also observe that d divides the two equal sides of the isosceles triangle: 10/5 = 2, 40/5 = 8, 104/13 = 8, 510/17 = 30, 1590/853 = 30, 21616/193 = 112, 100738/241 = 418, ....

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.
R. A. Johnson, Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
Eric Weisstein's World of Mathematics, Exradius
Eric Weisstein's World of Mathematics, Inradius

Cf. A185210, A208984, A210207, A231174, A350378.

nn=520;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a)(s-b)(s-c);If[area2>0&&IntegerQ[Sqrt[area2]]&&PrimeQ[Sqrt[a*b*c/(4*Sqrt[area2])*(a*b*c/(4*Sqrt[area2])-2*Sqrt[area2]/s)]],Print[Sqrt[area2]," ",c," ",b," ",a," ",Sqrt[area2]/s," ",a*b*c/(4*Sqrt[area2])," ",Sqrt[a*b*c/(4*Sqrt[area2])*(a*b*c/(4*Sqrt[area2])-2*Sqrt[area2]/s)]]]],{a,nn},{b,a},{c,b}]

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.

Original entry on oeis.org

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

Bernard Schott, Mar 11 2022

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.

			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.

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.

Cf. A231174, A350378, A350379, A352315.

More terms from Jinyuan Wang, Mar 12 2022

A352315 a(n) is the distance d between the incenter I and the circumcenter O of the integer-sided triangle whose sides correspond to the n-th primitive triple of A352314.

Original entry on oeis.org

5, 5, 21, 13, 91, 143, 95, 533, 221, 17, 407, 575, 341, 275, 703, 259, 377, 319, 53, 559, 4181, 793, 481, 3599, 715, 784, 943, 1955, 3965, 549, 7055, 6815, 2144, 1961

Offset: 1

Bernard Schott, Mar 11 2022

The triples of sides (a, b, c) are in increasing order of largest side c.
For the corresponding primitive triples and miscellaneous properties, formulas and references see A352314.
Two distinct such triangles can have the same distance OI (see examples).
From the table in A352314, when d is prime and the triangle ABC isosceles, then
-> d divides the two equal sides of this triangle, and also,
-> if d^2 = R, then r = (R-1)/2,
-> if d^2 = 3R then r = (R-3)/2.

			a(1) = 5 because with the smallest triple (10, 10, 16), we get 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 is an integer.
a(2) = 5 also because with the second triple (40, 40, 48), we get s = (40+40+48)/2 = 64, A = 768, r = 768/64 = 12, R = (40*40*48)/(4*768) = 25, and d = sqrt(25*(25-24)) = 5.
a(3) = 21 because with the third triple (16, 49, 55) that is the first triangle not isosceles, we get s = (16+49+55)/2 = 60, A = 220*sqrt(3), r = 11*sqrt(3)/3, R = (16*49*55)/(4*220*sqrt(3)) = 49*sqrt(3)/3, and d = sqrt(49^2/3 - (2*11*49)/3) = 21.

Cf. A231174, A350378, A350379, A352314, A352316.

lista(nn) = my(d, s); for(c=2, nn, for(b=1+c\2, c, for(a=1+c-b, b, s=(a+b+c)/2; if(denominator(d=a^2*b^2*c^2/16/s/(s-a)/(s-b)/(s-c)-a*b*c/2/s) == 1 && issquare(d) && gcd([a, b, c, d=sqrtint(d)]) == 1, print1(d, ", "))))); \\ Jinyuan Wang, Mar 15 2022

a(8) inserted by and a(12)-a(34) from Jinyuan Wang, Mar 15 2022

A352316 Perimeter of primitive integer-sided triangles such that the distance d = OI between the circumcenter O and the incenter I is also a positive integer.

Original entry on oeis.org

36, 128, 120, 288, 360, 500, 1008, 972, 1024, 1500, 1152, 1372, 2520, 3072, 2520, 3960, 4000, 4116, 4860, 5040, 4500, 5760, 6860, 5324, 6804, 6435, 8000, 7776, 8192, 10920, 8788, 9216, 10395, 10976

Offset: 1

Bernard Schott, Mar 14 2022

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.

			a(1) = 36 because the smallest triple is (10, 10, 16) with corresponding d = OI = A352315(1) = 5.

Cf. A231174, A350378, A350379, A352314, A352315.

a(n) = A352314(n, 1) + A352314(n, 2) + A352314(n, 3).

a(19)-a(34) from Jinyuan Wang, Mar 14 2022

cp's OEIS Frontend

A350379 Consider integer-sided triangles having integer area and let d be the distance between the center of the inscribed circle and the center of the circumscribed circle. The sequence lists the set of distances d which are prime numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A352315 a(n) is the distance d between the incenter I and the circumcenter O of the integer-sided triangle whose sides correspond to the n-th primitive triple of A352314.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A352316 Perimeter of primitive integer-sided triangles such that the distance d = OI between the circumcenter O and the incenter I is also a positive integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions