In $ riangle A B C$, if $A, B, C$ are in arithmetic progression, $\Delta=\frac{\sqrt{3}}{2}$ and $r_1 r_2=r_3 r$, then $R=$

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In $	riangle A B C$, if $A, B, C$ are in arithmetic progression, $\Delta=\frac{\sqrt{3}}{2}$ and $r_1 r_2=r_3 r$, then $R=$

Aicharya · Accepted Answer

$\begin{aligned} & A, B, C 	ext { are in AP, } r_1 r_2=r_3 r \ & \Rightarrow \frac{\Delta}{(s-a)} \cdot \frac{\Delta}{(s-b)}=\frac{\Delta}{(s-c)} \cdot \frac{\Delta}{s} \ & \Rightarrow \frac{(s-a)(s-b)}{s(s-c)}=1 \Rightarrow 	an ^2 \frac{C}{2}=1 \ & C=90^{\circ}, A+B+C=180^{\circ} \ & \Rightarrow 3 B=180^{\circ} \Rightarrow B=60^{\circ} \Rightarrow A=30^{\circ} \ & C=2 R, B C=2 R \sin 30^{\circ} \ & a=R, b=A C=2 R \sin 60^{\circ}=\sqrt{3} R \ & \Delta=\frac{a b}{2}=\frac{\sqrt{3} R^2}{2}=\frac{\sqrt{3}}{2} \Rightarrow R^2=1 \Rightarrow R=1\end{aligned}$

In $\triangle A B C$, if $A, B, C$ are in arithmetic progression, $\Delta=\frac{\sqrt{3}}{2}$ and $r_1…

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