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A350379 Consider integer-sided triangles having integer area and let d be the distance between the center of the inscribed circle and the center of the circumscribed circle. The sequence lists the set of distances d which are prime numbers.

%I A350379 #13 Feb 16 2025 08:34:02
%S A350379 5,13,17,53,193,241
%N A350379 Consider integer-sided triangles having integer area and let d be the distance between the center of the inscribed circle and the center of the circumscribed circle. The sequence lists the set of distances d which are prime numbers.
%C A350379 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.
%C A350379 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.
%C A350379 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.
%C A350379 The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.
%C A350379 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)).
%C A350379 +------------+--------+--------+---------+---------+---------+-----+
%C A350379 |         A  |     a  |     b  |     c   |    r    |    R    |  d  |
%C A350379 +------------+--------+--------+---------+---------+---------+-----+
%C A350379 |         48 |     10 |     10 |     16  |    8/3  |    25/3 |   5 |
%C A350379 |        768 |     40 |     40 |     48  |   12    |    25   |   5 |
%C A350379 |       3840 |     80 |    104 |    104  |   80/3  |   169/3 |  13 |
%C A350379 |     108000 |    480 |    510 |    510  |  144    |   289   |  17 |
%C A350379 |    1134000 |   1590 |   1590 |   1680  |  1400/3 |  2809/3 |  53 |
%C A350379 |  200202240 |  21280 |  21616 |  21616  | 18620/3 | 37249/3 | 193 |
%C A350379 | 4382077920 | 100320 | 100738 | 100738  | 29040   | 58081   | 241 |
%C A350379 ....................................................................
%C A350379 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, ....
%D A350379 Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.
%D A350379 R. A. Johnson, Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.
%H A350379 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/25678790">Solution of problem 125: Circumradius and Inradius</a>, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
%H A350379 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Exradius.html">Exradius</a>
%H A350379 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Inradius.html">Inradius</a>
%t A350379 nn=520;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a)(s-b)(s-c);If[area2>0&&IntegerQ[Sqrt[area2]]&&PrimeQ[Sqrt[a*b*c/(4*Sqrt[area2])*(a*b*c/(4*Sqrt[area2])-2*Sqrt[area2]/s)]],Print[Sqrt[area2]," ",c," ",b," ",a," ",Sqrt[area2]/s," ",a*b*c/(4*Sqrt[area2])," ",Sqrt[a*b*c/(4*Sqrt[area2])*(a*b*c/(4*Sqrt[area2])-2*Sqrt[area2]/s)]]]],{a,nn},{b,a},{c,b}]
%Y A350379 Cf. A185210, A208984, A210207, A231174, A350378.
%K A350379 nonn,more
%O A350379 1,1
%A A350379 _Michel Lagneau_, Dec 28 2021