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Newton's Formulas

Let a Triangle have side lengths $a$, $b$, and $c$ with opposite angles $A$, $B$, and $C$. Then

$\displaystyle {b+c\over a}$ $\textstyle =$ $\displaystyle {\cos[{\textstyle{1\over 2}}(B-C)]\over\sin({\textstyle{1\over 2}}A)}$  
$\displaystyle {c+a\over b}$ $\textstyle =$ $\displaystyle {\cos[{\textstyle{1\over 2}}(C-A)]\over\sin({\textstyle{1\over 2}}B)}$  
$\displaystyle {a+b\over c}$ $\textstyle =$ $\displaystyle {\cos[{\textstyle{1\over 2}}(A-B)]\over\sin({\textstyle{1\over 2}}C)}.$  

See also Mollweide's Formulas, Triangle


Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 146, 1987.

© 1996-9 Eric W. Weisstein