Question

Explain the Proof for Lemis Theorm

Answer 1

You mean Lami’s Theorem?

Consider the figure. Three concurrent forces net to static equilibrium at the origin. Let's assume for simplicity force a is along the positive x axis.

We have equilibrium when the components of force balance in both dimensions. In particular in the y dimension,

$b \sin C = c \sin B$

$\dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

We're free to move our coordinate system around to make any of the force vectors the x axis and then we'd get the same sine balance as in the Law of Sines for the remaining pair of sides.

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