<< just by chance, i was working on something related to this the other day. the formula for the circumradius of a convex cyclic quadrilateral, with side lengths a , b , c and d is:

/------------------------------------------------------------------ / (ab + cd) (ac + bd) (ad + bc) R = \ / ------------------------------------------------------------------ \/ (-a + b + c + d) (a - b + c + d) (a + b - c + d) (a + b + c - d)

the formula for the circumradius of a triangle can be obtained by putting d = 0 into this formula.

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This is a nice formula!

glad you like it; i thought it was pretty neat, too. perhaps i should have said that i derived this formula for what i was working on. i had not seen it before, though i can't imagine it's really "new". the symmetry in the variables, as you note, means that the circumradius is independent of the order of the sides. by exploiting this properly, it is possible to give a nice formula for the diagonal of a cyclic quadrilateral. i'll leave this as a cute puzzle for you. mike