In $\Delta ABC$ , if A,B,C are in A.P., then $\frac{a+c}{\sqrt{a^2-ac+c^2}}=$

$\begin{array}{l}2B=A+C\\ A+B+C=180\Rightarrow 3B=180\\ \Rightarrow B=60\\ \sin 2B=\sin (A+C)\end{array}$

$\begin{array}{l}2\sin B\cos B=\sin (A+C)\\ 2\sin (A+C)\cos B=\sin (A+C)\\ \cos B=\frac{1}{2}\\ \cos B=\cos 60\end{array}$

$\begin{array}{l}\frac{a^2+c^2-b^2}{2ac}=\frac{1}{2}\\ a^2+c^2-ac=b^2\\ b=\sqrt{a^2+c^2-ac}\end{array}$

$\frac{c+a}{b}=\frac{c+a}{\sqrt{c^2+a^2-ac}}=\frac{\cos \left(\frac{C-A}{2}\right)}{\sin \frac{B}{2}}$

$\frac{c+a}{\sqrt{c^2+a^2-ac}}=\frac{\frac{\cos C-A}{\sin 30}}{\sin 30}=2\cos \left(\frac{{\displaystyle A-C}}{{\displaystyle 2}}\right)$