Eliminate $x,y,z$ $;$
$ax+by+cz=ax^2+by^2+cz^2=xy+yz+xz=0 ;$
This question is from an olympiad book which does not provide solutions but answers to only selected questions this question is one of them
I have been trying this for long but didn't end up solving it
My approaches were as follows:- substituting $x$ from the last equation in the 1st and 2nd equation, then I tried to solve those two equations and the result I derived was
$a=(y+z)^3/(yz)^2 ;$
$b=(x+z)^3/(xz)^2 ;$
$c=(x+y)^3/(xy)^2 ;$
I don't know what can we further do to get the relation between $a,b,c$ and eliminate $x,y,z$
Secondly, it is obvious that
$(x+y+z)=(x^2+y^2+z^2) ;$
Then multiplying the equation by $a$ we can easily get $ax^2-ax$ and substituting that in the equation derived by subtracting 1st and 2nd equation but that doesn't help too
I am very confused about it....please give a solution to solve it ...
The answer given to the problem is:-
$(a+b+c)^3-4(a+b)(b+c)(c+a)+5abc=0 ;$
This is not a homework problem