In $ riangle A B C$, if $(a+c)^2=b^2+3 c a$, then $\frac{a+c}{2 R}=$

Question

In $	riangle A B C$, if $(a+c)^2=b^2+3 c a$, then $\frac{a+c}{2 R}=$

Aicharya · Accepted Answer

$\begin{aligned} & (a+c)^2=b^2+3 c a \Rightarrow a^2+c^2-b^2=c a \ & \cos B=\frac{a^2+c^2-b^2}{2 a c}=\frac{1}{2}=\cos 60^{\circ} \Rightarrow \frac{B}{2}=30^{\circ} \ & \frac{a+c}{2 R}=\sin A+\sin C=2 \sin \left(\frac{A+C}{2}ight) \cos \left(\frac{A-C}{2}ight) \ & =2 \sin \left(\frac{\pi}{2}-\frac{B}{2}ight) \cos \left(\frac{A-C}{2}ight) \ & =2 \cos \left(\frac{B}{2}ight) \cos \left(\frac{A-C}{2}ight) \ & =2 \cos 30^{\circ} \cos \left(\frac{A-C}{2}ight)=\sqrt{3} \cos \left(\frac{A-C}{2}ight)\end{aligned}$

In $\triangle A B C$, if $(a+c)^2=b^2+3 c a$, then $\frac{a+c}{2 R}=$

Solution