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

(In reply to re(4): Solution by Brian Smith)

Thanks for the explanation.

I noted the same about my expression.

 4*(r^2 - a^2)^2 + (2r^2 - b^2)^2 - 4*r^4 = 0

 4r^4 - 8a^2*r^2 + 4a^4 - 4b^2*r^2 + b^4 = 0

 4r^4 - 4(2a^2 + b^2)*r^2 + (2a^2 + b^2)^2 - 4a^2*b*2 = 0

 [2r^2 - (2a^2 + b^2)]^2 - 4a^2*b*2 = 0

 [(2r^2 - 2a^2 - b^2) - 2ab][(2r^2 - 2a^2 - b^2) + 2ab] = 0
 
  2[r^2 - (a + b/2)^2 - (b/2)^2)][(2r^2 - 2a^2 - b^2) + 2ab] = 0

   |     Your expression        |

Edited on April 25, 2007, 2:47 pm

Posted by Bractals on 2007-04-25 13:36:30