If $a, b, c$ are continued proportional, then $\left(\dfrac{a+b}{b+c}\right)^2$ equals—

For continued proportion $\dfrac{a}{b}=\dfrac{b}{c}=k$, write $a=bk$ and $b=ck$. Then $\dfrac{a+b}{b+c}=\dfrac{bk+ck}{... }=k$, and squaring gives $k^2$; likewise $\dfrac{a^2+b^2}{b^2+c^2}=k^2$. So the two expressions are equal.