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 the, relation to and/or relation:

    “You see, I am alive, I am alive
    I stand in good relation to the earth
    I stand in good relation to the gods
    I stand in good relation to all that is beautiful
    I stand in good relation to the daughter of Tsen-tainte
    You see, I am alive, I am alive”
    —N. Scott Momaday (b. 1934)

    “We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
    —D.H. (David Herbert)

    “Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
    —Honoré De Balzac (1799–1850)