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Given the length ot the sides $a , b$ and $c$ of $ \triangle ABC$.

What is the length of the radius of the circumcribed circle?

After some formula substitution I came to the monster formula:

$$ \frac {a b c}{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}} $$

Can this formula be simplified?

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  • $\begingroup$ Multiply and divide by $\sqrt{2}$, then write the denominator as sum of squares. $\endgroup$ – HarshCurious Sep 27 '14 at 7:22
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you will notice that the expression inside the square root sign can be factorized using the difference of two squares: $$ 2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4 = 4a^2b^2 - (a^2+b^2-c^2)^2 $$ $$ = (c^2-(a-b)^2)((a+b)^2-c^2) $$ $$ =(a+b+c)(a+b-c)(b+c-a)(c+a-b) $$ if you set s=$\frac{a+b+c}2$ then this expression becomes $$ 16s(s-a)(s-b)(s-c) $$ which you may recognize from a formula giving the area of a triangle in terms of the lengths of its three sides

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