Law of Sines/Proof 1

Theorem
For any triangle $\triangle ABC$:


 * $\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C}$

where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.

Proof
Construct the altitude from $B$.


 * Law Of Sines 1.png

It can be seen from the definition of sine that:
 * $\sin A = \dfrac h c$ and $\sin C = \dfrac h a$

Thus:
 * $h = c \sin A$ and $h = a \sin C$

This gives:
 * $c \sin A = a \sin C$

So:
 * $\dfrac a {\sin A} = \dfrac c {\sin C}$

Similarly, constructing the altitude from $A$ gives:
 * $\dfrac b {\sin B} = \dfrac c {\sin C}$