Question from higher algebra - hall knight

[smart_worker]

The Attempt at a Solution

basically i would rewrite the equation as,

(x/l)/(mb + nc - la) = (y/m)/(nc + la - mb) = (z/n)/(la + mb - nc)

upon adding,i get
(x/l + y/m + z/n)/(la+mb+nc)

but what they provided is,
(y/m + z/n)/2la = two similar expressions

==> (ny + mz)/a = (lz + nx)/b = (mx + ly)/c

i did not understand how did they get that.

since this is a proof problem i only used the "3. The Attempt at a Solution " section.

 

[smart_worker]

Image is the question.

 

[SammyS]

Image is the question.

As in:

 

[smart_worker]

As in:

ya this is the question sammy.I tried to solve this but couldn't.The explanation they provided couldn't be understood too

 

[lurflurf]

two facts are being derived and then used
$$k=\frac{a}{b}=\frac{c}{d}=\frac{e}{f}\rightarrow k=\frac{c+e}{d+f}=\frac{e+a}{f+b}=\frac{a+c}{b+d}$$
and
$$k=\frac{c+b}{d}=\frac{a+c}{e}=\frac{b+a}{f}\rightarrow k=\frac{2a}{e+f-d}=\frac{2b}{d+f-e}=\frac{2c}{d+e-f}$$

So we take the given equation put it in the right form and transform it the first way then we put it in the form needed and transform again. finally we put it in the form of the answer.