A350379 -id:A350379

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A350378

Integer areas of integer-sided triangles such that the distance d between the incenter and the circumcenter is a prime number.

+10
4

48, 768, 3840, 108000, 1134000, 200202240, 4382077920

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Subsequence of A231174. The sequence is probably infinite.

The corresponding sequence of the primes d is {5, 5, 13, 17, 53, 193, 241,...} (see A350379).

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 triangles: 10/5 = 2, 40/5 = 8, 104/13 = 8, 510/17 = 30, 1590/853 = 30, 21616/193 = 112, 100738/241 = 418,...}.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..7.

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

MATHEMATICA

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}]

CROSSREFS

Cf. A185210, A208984, A210207, A231174, A350379.

KEYWORD

nonn,more

AUTHOR

Michel Lagneau, Dec 28 2021

STATUS

approved

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
2

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,tabf

AUTHOR

Bernard Schott, Mar 11 2022

EXTENSIONS

More terms from Jinyuan Wang, Mar 12 2022

STATUS

approved

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.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..34.

EXAMPLE

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.

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Bernard Schott, Mar 11 2022

EXTENSIONS

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

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..34.

FORMULA

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

EXAMPLE

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

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Bernard Schott, Mar 14 2022

EXTENSIONS

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

STATUS

approved

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