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%I #24 Feb 06 2015 02:33:57
%S 276,342,1332,1638,1848,1884,2058,2094,2148,2268,2358,2424,2436,2760,
%T 2844,2856,2952,3108,3150,3276,3390,3624,3696,3810,4044,4056,3624,
%U 4512,4998,5112,5268,6192,6312,6786,6942,6984,7230,7374,7476,7962,7680,7722,8022,8712
%N Area of the Robbins pentagons.
%C See the first link and the table 2 page 23 for the results of the exploration of all pentagons with perimeter less than 400.
%C A Robbins pentagon is a cyclic polygon with 5 integer sides and integer area.
%C Any Robbins pentagon with five integer sides has integer area (proof in reference).
%C Theorem (Robbins). Consider a cyclic pentagon with sides a,b,c,d,e and area A. If s1, s2, s3, s4 and s5 are the symmetric polynomials in the squares of the sides, x = 16A^2, t=x-4*s2 + s1^2, u = 8*s3 + s1*t2, v = -64*s4 + t^2 and w = 128*s5, then u (hence the square of the area) satisfies the condition: x*v^3 + u^2 * v^2 - 18*x*u*v*w - 27*x^2*w^2.
%H Ralph H. Buchholz and James A. MacDougall, <a href="http://dx.doi.org/10.1016/j.jnt.2007.05.005">Cyclic polygons with rational Sides and Area</a>, Journal of Number Theory, Volume 128, Issue 1, January 2008, Pages 17-48.
%H Kival Ngaokrajang, <a href="/A228517/a228517.pdf">Illustration for n = 1..6</a>
%H D. P. Robbins, <a href="http://www.jstor.org/stable/2974766">Areas of polygons inscribed in a circle</a>, Amer. Math. Monthly, 102 (1995), 523-530.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclicPentagon.html">Cyclic Pentagon</a>
%e 276 is the area of the pentagon with sides (7, 7, 15, 15, 24).
%K nonn
%O 1,1
%A _Michel Lagneau_, Aug 24 2013