A232274 Integer areas A of the integer-sided triangles such that the length of the inradius and the circumradius are both a perfect square.

168, 2688, 13608, 43008, 105000, 108000, 217728, 403368, 688128, 1102248, 1680000, 1728000

Offset: 1

Author

Michel Lagneau, Nov 22 2013

Keywords

• nonn
• hard

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 |

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

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A188158, A208984.

Programs

Mathematica

nn=2000;Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a)(s-b)(s-c);If[0