If $ riangle A B C$ is right angled at $A$, then $r_2+r_3$ is equal to

Question

If $	riangle A B C$ is right angled at $A$, then $r_2+r_3$ is equal to

Aicharya · Accepted Answer

Given that $\angle A=90^{\circ}$ Now, $ r_2+r_3=4 R \cos \frac{A}{2} \sin \frac{B}{2} \cos \frac{C}{2}+4 R \cos \frac{A}{2} $ $ \begin{aligned} & =4 R \cos \frac{A}{2}\left[\sin \frac{B}{2} \cos \frac{C}{2}+\cos \frac{B}{2} \sin \frac{C}{2}ight] \ & =4 R \cos 45^{\circ} \cdot \sin \left(\frac{B+C}{2}ight) \ & =4 R \cdot \frac{B}{\sqrt{2}} \cdot \sin \frac{C}{2}\left(\frac{\pi}{2}-\frac{A}{2}ight) \ & =4 R \cdot \frac{1}{\sqrt{2}} \cdot \cos \left(\frac{A}{2}ight) \ & =4 R \cdot \frac{1}{\sqrt{2}} \cdot \cos 45^{\circ} \ & =4 R \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}=2 R \ 	herefore r_2 & +r_3=2 R \end{aligned} $ Also, $ \begin{aligned} 	ext { Also, } & & r_1+r_2+r_3-r=4 R=2(2 R) \ & & r_1+r_2+r_3-r=2\left(r_2+r_3ight) \ \Rightarrow & & r_2+r_3=r_1-r \end{aligned} $

If $\triangle A B C$ is right angled at $A$, then $r_2+r_3$ is equal to

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