The corresponding integer areas of integer-sided triangles such that the distance between the incenter and the circumcenter is a prime number is given by the sequence A350378.
In geometry, Euler's theorem states that the distance between the incenter and circumcenter can be expressed as d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius.
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.
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 inradius, R is the circumradius and d is the distance between the incenter and circumcenter with d = sqrt(R(R-2r)).
+------------+--------+--------+---------+---------+---------+-----+
| A | a | b | c | r | R | d |
+------------+--------+--------+---------+---------+---------+-----+
| 48 | 10 | 10 | 16 | 8/3 | 25/3 | 5 |
| 768 | 40 | 40 | 48 | 12 | 25 | 5 |
| 3840 | 80 | 104 | 104 | 80/3 | 169/3 | 13 |
| 108000 | 480 | 510 | 510 | 144 | 289 | 17 |
| 1134000 | 1590 | 1590 | 1680 | 1400/3 | 2809/3 | 53 |
| 200202240 | 21280 | 21616 | 21616 | 18620/3 | 37249/3 | 193 |
| 4382077920 | 100320 | 100738 | 100738 | 29040 | 58081 | 241 |
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From the previous table, we observe that the triangles are isosceles, the distance between the incenter and the circumcenter is d = sqrt(R) if R is a perfect square, or d = sqrt(3R) if R is of the form k^2/3, k integer. We also observe that d divides the two equal sides of the isosceles triangle: 10/5 = 2, 40/5 = 8, 104/13 = 8, 510/17 = 30, 1590/853 = 30, 21616/193 = 112, 100738/241 = 418,...}.
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