If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}$, then:
$\textbf{(A) }a \text{ must equal }c\qquad\textbf{(B) }a+b+c+d\text{ must equal zero}\qquad$
$\textbf{(C) }\text{either }a=c\text{ or }a+b+c+d=0\text{, or both}\qquad$
$\textbf{(D) }a+b+c+d\ne 0\text{ if }a=c\qquad$
$\textbf{(E) }a(b+c+d)=c(a+b+d)$
Cross-multiplying gives:
$(a+b)(a+d) = (b+c)(c+d)$
$a^2 + ad + ab + bd = bc + bd + c^2 + cd$
$a^2 + ad + ab   - bc - c^2 - cd = 0$
$a(a + b + d) - c(b+c+d)= 0$
This looks close to turning into option C, but we don't have a $c$ term in the first parentheses, and we don't have an $a$ term in the second parentheses to allow us to complete the factorization.  However, if we both add $ac$ and subtract $ac$ on the LHS, we get:
$a(a + b + d) + ac - c(b+c+d) - ca= 0$
$a(a+b+d +c) - c(b+c+d+a) = 0$
$(a-c)(a+b+c+d) = 0$
This is equivalent to $\boxed{\textbf{(C)}}$