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A233317
Integer areas A of the integer-sided triangles such that the inradius and the radius of the three excircles are perfect squares.
0
108, 1728, 8748, 12348, 27648, 67500, 139968, 197568, 259308, 442368, 707472, 708588, 1000188, 1080000, 1581228, 2239488, 3084588, 3161088, 4148928, 5467500, 7077888, 7717500, 9020268, 11319552, 11337408, 14074668, 16003008, 17280000, 21003948, 25299648
OFFSET
1,1
COMMENTS
Subset of A185210.
A = sqrt(s*(p-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);
The inradius is r = A/s;
The radii of the three excircles are r1 = 2*A/(-a+b+c); x2 = 2*A*b/(a-b+c); x3 = 2*A*c/(a+b-c).
The areas A of the primitive triangles of sides (a,b,c) are 108, 12348, ...
The areas of the nonprimitive triangles of sides (a*p^2, b*p^2, c*p^2) are in the sequence with the value A*p^4.
The following table gives the first values (A, a, b, c, r, r1, r2, r3) where A is the area of the triangle, a, b, c the integer sides, r, r1, r2 and r3 are respectively the length of the inradius and the radius of the three excircles.
+--------+-----+------+------+------+------+------+------+
| A | a | b | c | r | r1 | r2 | r3 |
+--------+-----+------+------+------+------+------+------+
| 108 | 15 | 15 | 24 | 2^2 | 3^2 | 3^2 | 6^2 |
| 1728 | 60 | 60 | 96 | 4^2 | 6^2 | 6^2 | 12^2 |
| 8748 | 135 | 135 | 216 | 6^2 | 9^2 | 9^2 | 18^2 |
| 12348 | 91 | 280 | 315 | 6^2 | 7^2 | 14^2 | 21^2 |
| 27648 | 240 | 240 | 384 | 8^2 | 12^2 | 12^2 | 24^2 |
| 67500 | 375 | 375 | 600 | 10^2 | 15^2 | 15^2 | 30^2 |
| 139968 | 540 | 540 | 864 | 12^2 | 18^2 | 18^2 | 36^2 |
| 197568 | 364 | 1120 | 1260 | 12^2 | 14^2 | 28^2 | 42^2 |
| 259308 | 735 | 735 | 1176 | 14^2 | 21^2 | 21^2 | 42^2 |
+--------+-----+------+------+------+------+------+------+
LINKS
Eric Weisstein's World of Mathematics, Excircles.
Eric Weisstein's World of Mathematics, Exradius.
Eric Weisstein's World of Mathematics, Inradius.
MATHEMATICA
nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[area2>0&& IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[Sqrt[area2]/s]] && IntegerQ[Sqrt[2*Sqrt[area2]/(-a+b+c)]] && IntegerQ[Sqrt[2*Sqrt[area2]/(a-b+c)]] && IntegerQ[Sqrt[2*Sqrt[area2]/(a+b-c)]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
CROSSREFS
Cf. A185210.
Sequence in context: A115135 A202309 A279981 * A263970 A250339 A263961
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 07 2013
STATUS
approved