OFFSET
1,1
COMMENTS
See the first link and the table 2 page 23 for the results of the exploration of all pentagons with perimeter less than 400.
A Robbins pentagon is a cyclic polygon with 5 integer sides and integer area.
Any Robbins pentagon with five integer sides has integer area (proof in reference).
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.
LINKS
Ralph H. Buchholz and James A. MacDougall, Cyclic polygons with rational Sides and Area, Journal of Number Theory, Volume 128, Issue 1, January 2008, Pages 17-48.
Kival Ngaokrajang, Illustration for n = 1..6
D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.
Eric Weisstein's World of Mathematics, Cyclic Pentagon
EXAMPLE
276 is the area of the pentagon with sides (7, 7, 15, 15, 24).
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Michel Lagneau, Aug 24 2013
STATUS
approved