The corresponding integer areas of integersided 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(R2r)), 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(sa)(sb)(sc)), 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(R2r)).
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 A  a  b  c  r  R  d 
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 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,...}.
