Proving a=b=c in a Ratio Proportion Problem

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Homework Statement

if(i have attached the equation's pic) then
prove a=b=c or (a+b+c)(x+y+z)=ax+by+cz

Homework Equations

if a/b=c/d=e/f
each ratio = (a+c+e)/(b+d+f)

The Attempt at a Solution

I was able to prove (a+b+c)(x+y+z)=ax+by+cz
like this:
each ratio= (a²+b²+c²-ab-bc-ca)/x+y+z
also
each ratio= (a³+b³+c[SUP3[/SUP]-3abc)/ax+by+cz
(i hope i could make you understand how i did this)
since
(a²+b²+c²-ab-bc-ca)/x+y+z =(a³+b³+c[SUP3[/SUP]-3abc)/ax+by+cz
→(a²+b²+c²-ab-bc-ca)/x+y+z=
(a+b+c)(a²+b²+c²-ab-bc-ca)/ax+by+cz
therefore
(a+b+c)(x+y+z)=ax+by+cz
fine uptill here but i can't understand how do i prove a=b=c??
tried it like this
(a+b+c)(x+y+z)=ax+by+cz
→bx+cx+ay+cy+az+bz=0

i can go further than this .Please give me a clue .
And if i have gone wrong somewhere please point it out.
Thank you.

 

[SammyS]

No picture is attached.

 

[HallsofIvy]

Well, the picture is attached but there is no equation in it- just a sum of three expressions. What was this supposed to be equal to?