In a triangle ABC,D, E,and F are the points o | vedclass

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(D) Let $s$ be the semi-perimeter of $	riangle ABC$. The lengths of the tangents from the vertices to the incircle are given by $\alpha = s-a, \beta

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$r^2$

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(D) Let $s$ be the semi-perimeter of $	riangle ABC$. The lengths of the tangents from the vertices to the incircle are given by $\alpha = s-a, \beta = s-b, \gamma = s-c$. Summing these,we get $\alpha+\beta+\gamma = 3s - (a+b+c) = 3s - 2s = s$. The area of the triangle $S$ is given by Heron's formula: $S = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{s \alpha \beta \gamma}$. We also know that $S = rs$,where $r$ is the inradius. Thus,$r^2 s^2 = s \alpha \beta \gamma$,which simplifies to $r^2 s = \alpha \beta \gamma$. Substituting $s = \alpha+\beta+\gamma$,we get $r^2 = \frac{\alpha \beta \gamma}{\alpha+\beta+\gamma}$.

In a $\triangle ABC$,$D, E$,and $F$ are the points of contact of the incircle with the sides $AB, BC$,and $CA$ respectively,such that $AD=\alpha, BE=\beta$,and $CF=\gamma$. Then $\frac{\alpha \beta \gamma}{\alpha+\beta+\gamma} = $

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