|
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 |
+--------+-----+------+------+------+------+------+------+
| 