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?

  • $\begingroup$ Multiply and divide by $\sqrt{2}$, then write the denominator as sum of squares. $\endgroup$ – HarshCurious Sep 27 '14 at 7:22

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.