Triangle Law And Polygon Law

Sine rule:
In any triangle the length of the side is proportional to the sine of the angle opposite to it. (or) \( \frac{a} {{\sin A}} = \frac{b} {{\sin B}} = \frac{c} {{\sin C}} \)    
                                

Proof: In \( \Delta ABD,\sin A = \frac{{h_1 }} {c} \Rightarrow h_1 = cSinA \) .......... (1)
    In \( \Delta BDC,\sin C = \frac{{h_1 }} {a} \Rightarrow h_1 = aSinC \) .............(2)
    From (1) and (2) c sin A = a sin C (or)  \( \frac{c} {{\sin C}} = \frac{a} {{\sin A\,}}\,\,\,\,....................(3) \)   
    From \( \Delta \,ACE,\,\,\sin C = \frac{{h_2 }} {b} \Rightarrow h_2 = b\,\sin C \) .........(4)
         \( \Delta \,ABE,\,\,\sin (180^0 - B) = \frac{{h_2 }} {C} \Rightarrow h_2 = c\,\sin \left( {180^0 - B} \right) = c\,\sin B \)
         \( \Rightarrow h_2 = c\,\sin B \)   ................................ (5)
    From (4) and (5)  b sin C = c sin B (or)  \( \frac{b} {{\sin B}} = \frac{c} {{\sin C}}\,\,\,\,....................\left( 6 \right) \)  
    from (3) and (6)  \( \frac{a} {{\sin A}} = \frac{b} {{\sin B}} = \frac{c} {{\sin C}} \)