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The distance between the incenter and circumcenter is given by d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius, a result known as the Euler triangle formula (see the link below).
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.
Properties of this sequence:
It appears that the triangles are isosceles.
a(n) = 48*m where the integers m are not squarefree: {m} ={1, 4, 9, 16, 25, 36, 49, 64, 80, 81, 100, 121, 125, 144, 169, 196, 225, 256, 270, 289, ...}, and the areas of the primitive triangles are 48, 3840, 6000, ... The integers m are not squarefree.
The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
The subsequence of the areas of triangles with inradius, circumradius and distance between the incenter and circumcenter integers is {432, 1728, 3072, 3888, 6000, 6912, ...}.
The following table gives the first values (A, a, b, c, R, r, d) where A is the area of the triangles, a, b, c are the integer sides of the triangles, R is the circumradius, r is the inradius and d is the distance between the incenter and circumcenter:
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| A | a | b | c | R | r | d |
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| 48 | 10 | 10 | 16 | 25/3 | 8/3 | 5 |
| 192 | 20 | 20 | 32 | 50/3 | 16/3 | 10 |
| 432 | 30 | 30 | 48 | 25 | 8 | 15 |
| 768 | 40 | 40 | 64 | 100/3| 32/3 | 20 |
| 1200 | 50 | 50 | 80 | 125/3| 40/3 | 25 |
| 1728 | 60 | 60 | 96 | 50 | 16 | 30 |
| 2352 | 70 | 70 | 112 | 175/3| 56/3 | 35 |
| 3072 | 80 | 80 | 96 | 50 | 24 | 10 |
| 3072 | 80 | 80 | 128 | 200/3| 64/3 | 40 |
| 3840 | 80 | 104 | 104 | 169/3| 80/3 | 13 |
| 3888 | 90 | 90 | 144 | 75 | 24 | 45 |
| 4800 |100 | 100 | 160 | 250/3| 80/3 | 50 |
| 5808 |110 | 110 | 176 | 275/3| 88/3 | 55 |
| 6000 |130 | 130 | 240 | 169 | 24 |143 |
| 6912 |120 | 120 | 144 | 75 | 36 | 15 |
| 6912 |120 | 120 | 192 | 100 | 32 | 60 |
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