A230491 Integer areas of the integer-sided triangles such that the length of the inradius is a square.

6, 84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 432, 468, 486, 504, 540, 594, 630, 684, 720, 756, 864, 990, 1026, 1080, 1116, 1134, 1152, 1224, 1332, 1344, 1404, 1440, 1494, 1536, 1584, 1638, 1680, 1710, 1728, 1782, 1824, 1872, 1890

Offset: 1

Author

Michel Lagneau, Oct 20 2013

Keywords

• nonn

Comments

The primitive areas are 6, 84, 108, 120, 132, 144, 156, 168, ...

The non-primitive areas 16*a(n) are in the sequence because if r is the inradius corresponding to a(n), then 4*r is the inradius corresponding to 16*a(n).

The following table gives the first values (A, r, a, b, c) where A is the integer area, r the inradius and a, b, c are the integer sides of the triangle.

******************************

* A * r * a * b * c *

*******************************

* 6 * 1 * 3 * 4 * 5 *

* 84 * 4 * 13 * 14 * 15 *

* 96 * 4 * 12 * 16 * 20 *

* 108 * 4 * 15 * 15 * 24 *

* 120 * 4 * 10 * 24 * 26 *

* 132 * 4 * 11 * 25 * 30 *

* 144 * 4 * 18 * 20 * 34 *

* 156 * 4 * 15 * 26 * 37 *

* 168 * 4 * 10 * 35 * 39 *

* 180 * 4 * 9 * 40 * 41 *

* 240 * 4 * 12 * 50 * 58 *

* 264 * 4 * 33 * 34 * 65 *

* 300 * 4 * 25 * 51 * 74 *

* 324 * 4 * 9 * 75 * 78 *

* 396 * 4 * 11 * 90 * 97 *

* 420 * 4 * 21 * 85 * 104 *

* 432 * 9 * 30 * 30 * 36 *

* 468 * 9 * 25 * 39 * 40 *

.........................

Examples

			84 is in the sequence because the area of triangle (13, 14, 15) is given by Heron's formula A = sqrt(21*(21-13)*(21-14)*(21-15))= 84 where the number 21 is the semiperimeter and the inradius is given by r = A/s = 84/21 = 4 is a square.
		

References

• Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

• Eric W. Weisstein, MathWorld: Inradius

Crossrefs

Cf. A000290, A188158, A228383.

Programs

Mathematica

nn = 600; lst = {}; 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]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Formula

Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula) and inradius r = A/s.