<s>[INST] If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then.. If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then
(A) $a=c$
(B) either $a=c$ or $a+b+c+d=0$
(C) $a+b+c+d=0$
(D) $a=c$ and $b=d$
I solved $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ and got $a(a+b+d)=c(c+b+d)$ and so I thought that (A) is the correct option. But the correct answer is (B).
I'm how $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ if $a+b+c+d=0$. Please help.
 [/INST] You made a small error when you solved the equation.  You should have gotten
$$a(a+b+d)=c(c+b+d)$$
Note the $c$ in $c+b+d$ on the right.  Can you take it from there?
Added after the OP's edit:  Now write the above as
$$a((a+b+c+d)-c)=c((a+b+c+d)-a)$$
which becomes
$$(a-c)(a+b+c+d)=0$$
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