A230479 Integer areas of the integer-sided triangles such that the length of the circumradius is a square.

168, 336, 432, 600, 768, 2688, 5376, 6000, 6912, 9600, 12288, 13608, 14280, 20280, 27216, 28560, 30720, 32928, 34560, 34992, 38640, 43008, 46200, 48600, 62208, 69360, 77280, 86016, 96000, 105000, 108000, 110592, 118272, 153600, 196608

Offset: 1

Author

Michel Lagneau, Oct 20 2013

Keywords

• nonn

Comments

The primitive areas are 168, 338, 432, 600, 768, 13608, 14280, 20280, 27216, ...

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

Each circumradius belongs to the sequence {25, 100, 169, 225, 289, 400, 625, 676, ...}, and it seems that this last sequence is A198385 (second of a triple of squares in arithmetic progression).

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

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

* A * R * a * b * c *

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

* 168 * 25 * 14 * 30 * 40 *

* 336 * 25 * 14 * 48 * 50 *

* 432 * 25 * 30 * 30 * 48 *

* 600 * 25 * 30 * 40 * 50 *

* 768 * 25 * 40 * 40 * 48 *

* 2688 * 100 * 56 * 120 * 160 *

* 5376 * 100 * 56 * 192 * 200 *

* 6912 * 100 * 120 * 120 * 192 *

* 9600 * 100 * 120 * 160 * 200 *

* 12288 * 100 * 160 * 160 * 192 *

* 13608 * 225 * 126 * 270 * 360 *

* 14280 * 169 * 130 * 238 * 312 *

* 20280 * 169 * 130 * 312 * 338 *

* 27216 * 225 * 126 * 432 * 450 *

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

Examples

			168 is in the sequence because the area of the triangle (14, 30, 40) is given by Heron's formula A = sqrt(42*(42-14)*(42-30)*(42-40))= 168 where the number 42 is the semiperimeter, and the circumcircle is given by R = a*b*c/(4*A) = 14*30*40/(4*168) = 25, which 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: Circumradius

Crossrefs

Cf. A188158, A208984, A210207.

Programs

Mathematica

nn = 1000; 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[a*b*c/(4*Sqrt[area2])]], 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);

Circumradius R = a*b*c/4A.