If $$\frac{a+b}{b+c} = \frac{c+d}{d+a}$$, which of the following statements is always true?

Given, $$\dfrac{a+b}{b+c} = \dfrac{c+d}{d+a}$$

From, the properties of ratio, $$\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}=\dfrac{a+b+c+d}{a+b+c+d}$$

Now there are two possibilities,

i.) $$a+b+c+d\ne\ 0$$

When, $$a+b+c+d\ne\ 0$$, $$\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}=\dfrac{a+b+c+d}{a+b+c+d}=1$$

So, $$\dfrac{a+b}{b+c}=1$$

or, $$a+b=b+c$$

or, $$a=c$$

ii.) $$a+b+c+d=\ 0$$

In this case $$\dfrac{a+b+c+d}{a+b+c+d}$$ won't give a finite value.

So, either $$a=c$$ or $$a+b+c+d=0$$ is always true.