In triangle ABC, find the value of r_1+r_2+r_ | vedclass

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(B) We know that the exradii of a triangle are given by $r_1 = \frac{\Delta}{s-a}$, $r_2 = \frac{\Delta}{s-b}$, and $r_3 = \frac{\Delta}{s-c}$.Summing

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$4R+r$

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(B) We know that the exradii of a triangle are given by $r_1 = \frac{\Delta}{s-a}$, $r_2 = \frac{\Delta}{s-b}$, and $r_3 = \frac{\Delta}{s-c}$.Summing these, we get $r_1+r_2+r_3 = \Delta \left( \frac{1}{s-a} + \frac{1}{s-b} + \frac{1}{s-c} \right)$.Using the identity $\frac{1}{s-a} + \frac{1}{s-b} + \frac{1}{s-c} - \frac{1}{s} = \frac{4R}{r} \cdot \frac{1}{s}$ is not standard, but we know $r_1+r_2+r_3 = 4R+r$.Alternatively, using the property $r_1+r_2+r_3 = 4R+r$ where $R$ is the circumradius and $r$ is the inradius, the result is $4R+r$.

In $\triangle ABC$, find the value of $r_1+r_2+r_3$.

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