Consider the cyclic quadrilateral with sides 1, 4, 8, 7 in that order. What is its circumdiameter?
The area of a cyclic quadrilateral can be calculated using A^2 = (s-a)*(s-b)*(s-c)*(s-d) where s is the semiperimeter.
An alternative area formula involves the circumradius, A^2 = (a*b+c*d)*(a*c+b*d)*(a*d+b*c)/(16r^2).
From the values give, s = (1+4+7+8)/2 = 10.
Then A^2 = (10-1)*(10-4)*(10-8)*(10-7) = 324
And A^2 = (1*4+8*7)*(1*8+4*7)*(1*7+4*8)/(16r^2) = 5265/r^2
Equate the two area results to yield 324 = 5265/r^2, which implies the circumradius is sqrt(65)/2, which means the circumdiameter sought is sqrt(65)
Edited to fix error 5625->5265
Edited on March 17, 2019, 12:55 pm
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