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 A Near Diophantine Octagon Problem (Posted on 2007-04-22)
The cyclic octagon ABCDEFGH has the sides a√2, a√2, a√2, a√2, b, b, b and b respectively in that order. Each of a, b and r are positive integers, where r is the radius of the circumcircle.

Analytically determine:

(i) The minimum value of a with a < b

(ii) The minimum value of b with a > b

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (2 votes)

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 Analytic/Synthetic Solution | Comment 1 of 10
`Let alpha and beta be the central anglessubtended by sides a*sqrt(2) and b respectively. From the law of cosines we have`
`                r^2 - a^2  cos(alpha) = -----------                   r^2`
`               2r^2 - b^2  cos(beta) = ------------                   2r^2`
`Since the sum of the central angles of anoctagon is 360 degrees, we have`
`  4*alpha + 4*beta = 360`
`         or`
`  alpha + beta = 90`
`Therefore,`
`  cos(alpha)^2 + cos(beta)^2 = 1`
`               or`
`    r^2 - a^2         2r^2 - b^2  [-----------]^2 + [------------]^2 = 1       r^2                2r^2`
`Thus, the integers a, b, and r must satisfythe following constraints:`
`      0 < a < r        0 < b < r*sqrt(2)`
`      4*[r^2 - a^2]^2 + [2r^2 - b^2]^2 - 4*r^4 = 0`
`From these, I do not know how to find the integersanalytically. Using Perl I was able to find the following answers:`
`For a < b,    (a,b,r) = (1,6,5)`
`For b < a,    (a,b,r) = (17,14,25)`
` `

 Posted by Bractals on 2007-04-22 14:36:52

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