Let alpha and beta be the central angles
subtended 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 an
octagon 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 satisfy
the 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 integers
analytically. 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 20070422 14:36:52 