Law of Sines - Relation To The Circumcircle

Relation To The Circumcircle

In the identity

the common value of the three fractions is actually the diameter of the triangle's circumcircle. It can be shown that this quantity is equal to

\begin{align} \frac{abc} {2S} & {} = \frac{abc} {2\sqrt{s(s-a)(s-b)(s-c)}} \\ & {} = \frac {2abc} {\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4) }}, \end{align}

where S is the area of the triangle and s is the semiperimeter

The second equality above is essentially Heron's formula.

Read more about this topic:  Law Of Sines

Famous quotes containing the words relation to and/or relation:

    “Whoever has a keen eye for profits, is blind in relation to his craft.”
    —Sophocles (497–406/5 B.C.)

    “The adolescent does not develop her identity and individuality by moving outside her family. She is not triggered by some magic unconscious dynamic whereby she rejects her family in favour of her peers or of a larger society.... She continues to develop in relation to her parents. Her mother continues to have more influence over her than either her father or her friends.”
    —Terri Apter (20th century)