Hello rana mohamed
Welcome to Math Help Forum! Originally Posted by
rana mohamed there are more than three ways to solve this proof , but i don't know how to solve it
a triangle m n l ,prove that :
1/n sin L + 1/l sin M + 1/m sin N =K/A
where K is half of the triangle's perimeter and A is the triangle's area
I think you want to prove
$\displaystyle \frac{1}{n \sin L}+\frac{1}{l \sin M}+ \frac{1}{m \sin N} = \frac{K}{A}$
Note that $\displaystyle A = \tfrac12mn\sin L= \tfrac12ln\sin M= \tfrac12lm\sin N$
$\displaystyle \Rightarrow \sin L = \frac{2A}{mn}$, etc
$\displaystyle \Rightarrow \frac{1}{n \sin L}+\frac{1}{l \sin M}+ \frac{1}{m \sin N}$
$\displaystyle =\frac{mn}{2An}+\frac{ln}{2Al}+\frac{lm}{2Am}$
$\displaystyle =\frac{m}{2A}+\frac{n}{2A}+\frac{l}{2A}$
$\displaystyle =\frac{l+m+n}{2A}$
$\displaystyle =\frac{k}{A}$
Grandad