

A228517


Area of the Robbins pentagons.


1



276, 342, 1332, 1638, 1848, 1884, 2058, 2094, 2148, 2268, 2358, 2424, 2436, 2760, 2844, 2856, 2952, 3108, 3150, 3276, 3390, 3624, 3696, 3810, 4044, 4056, 3624, 4512, 4998, 5112, 5268, 6192, 6312, 6786, 6942, 6984, 7230, 7374, 7476, 7962, 7680, 7722, 8022, 8712
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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=x4*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



EXAMPLE

276 is the area of the pentagon with sides (7, 7, 15, 15, 24).


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



