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%I #11 Jan 24 2022 15:42:23
%S 48,768,3840,108000,1134000,200202240,4382077920
%N Integer areas of integer-sided triangles such that the distance d between the incenter and the circumcenter is a prime number.
%C Subsequence of A231174. The sequence is probably infinite.
%C The corresponding sequence of the primes d is {5, 5, 13, 17, 53, 193, 241,...} (see A350379).
%C 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 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 The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.
%C 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 +------------+--------+--------+---------+---------+---------+-----+
%C | A | a | b | c | r | R | d |
%C +------------+--------+--------+---------+---------+---------+-----+
%C | 48 | 10 | 10 | 16 | 8/3 | 25/3 | 5 |
%C | 768 | 40 | 40 | 48 | 12 | 25 | 5 |
%C | 3840 | 80 | 104 | 104 | 80/3 | 169/3 | 13 |
%C | 108000 | 480 | 510 | 510 | 144 | 289 | 17 |
%C | 1134000 | 1590 | 1590 | 1680 | 1400/3 | 2809/3 | 53 |
%C | 200202240 | 21280 | 21616 | 21616 | 18620/3 | 37249/3 | 193 |
%C | 4382077920 | 100320 | 100738 | 100738 | 29040 | 58081 | 241 |
%C ....................................................................
%C 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 triangles: 10/5 = 2, 40/5 = 8, 104/13 = 8, 510/17 = 30, 1590/853 = 30, 21616/193 = 112, 100738/241 = 418,...}.
%D Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.
%D R. A. Johnson, Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Exradius.html">Exradius</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Inradius.html">Inradius</a>
%t 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 Cf. A185210, A208984, A210207, A231174, A350379.
%K nonn,more
%O 1,1
%A _Michel Lagneau_, Dec 28 2021