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A232274 Integer areas A of the integer-sided triangles such that the length of the inradius and the circumradius are both a perfect square. 1
168, 2688, 13608, 43008, 105000, 108000, 217728, 403368, 688128, 1102248, 1680000, 1728000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subset of A208984.

The areas of the primitive triangles of sides (a, b, c) and inradius, circumradius equals respectively to r and R are 672, 108000, ...  The sides of the nonprimitive triangles are of the form (a*k^2, b*k^2, c*k^2) with r' = r*k^2 and R' = R*k^2 where r', R' are respectively the inradius and the circumradius of the nonprimitive triangles. The areas A' of the nonprimitive triangles are A' = A*k^4.

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).

+---------+------+------+------+-----+------+

|    A    |   a  |   b  |   c  |  r  |   R  |

+---------+------+------+------+-----+------+

|     168 |   14 |   30 |   40 |   4 |   25 |

|    2688 |   56 |  120 |  160 |  16 |  100 |

|   13608 |  126 |  270 |  360 |  36 |  225 |

|   43008 |  224 |  480 |  640 |  64 |  400 |

|  105000 |  350 |  750 | 1000 | 100 |  625 |

|  108000 |  480 |  510 |  510 | 144 |  289 |

|  217728 |  504 | 1080 | 1440 | 144 |  900 |

|  403368 |  686 | 1470 | 1960 | 196 | 1225 |

|  688128 |  896 | 1920 | 2560 | 256 | 1600 |

| 1102248 | 1134 | 2430 | 3240 | 324 | 2025 |

| 1680000 | 1400 | 3000 | 4000 | 400 | 2500 |

| 1728000 | 1920 | 2040 | 2040 | 576 | 1156 |

+---------+------+------+------+-----+------+

LINKS

Table of n, a(n) for n=1..12.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.

Eric W. Weisstein, MathWorld: Circumradius

Eric W. Weisstein, MathWorld: Inradius

EXAMPLE

a(1) = 168 because, for (a,b,c) = (14, 30, 40) => s= (14 + 30 + 40)/2 = 42, and

A = sqrt(42*(42-14)*(42-30)*(42-40)) = sqrt(28224) = 168;

R = abc/4A = 14*30*40/(4*168) = 25;

r = A/s = 168/42 = 4.

MATHEMATICA

nn=2000; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a)(s-b)(s-c); If[0<area2&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[Sqrt[area2]/s]]&&IntegerQ[Sqrt[a*b*c/(4*Sqrt[area2])]], Print[Sqrt[area2], " ", a " ", b, " ", c, " ", Sqrt[area2]/s, " ", a*b*c/(4*Sqrt[area2])]]], {a, nn}, {b, a}, {c, b}]

CROSSREFS

Cf. A188158, A208984.

Sequence in context: A279725 A234730 A234815 * A110075 A331908 A231995

Adjacent sequences:  A232271 A232272 A232273 * A232275 A232276 A232277

KEYWORD

nonn,hard

AUTHOR

Michel Lagneau, Nov 22 2013

STATUS

approved

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Last modified January 20 21:46 EST 2022. Contains 350472 sequences. (Running on oeis4.)