In $Δ A B C, r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} =$

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$16 R^{2} - (a^{2} + b^{2} + c^{2})$

$16 R^{2} - (a^{2} + b^{2} - c^{2})$

$16 R^{2} - (a^{2} - b^{2} + c^{2})$

$16 R^{2} - (a^{2} - b^{2} - c^{2})$

answer is A.

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Detailed Solution

$\begin{array}{l} Δ A B C, r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} = \\ r_{1} + r_{2} + r_{3} - r = 4 R \\ {(r_{1} + r_{2} + r_{3} - r)}^{2} = 16 R^{2} \end{array}$

$\begin{array}{l} \Rightarrow r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} + 2 (r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}) \\ \Rightarrow r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} + 2 (r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}) \\ - 2 r (r_{1} + r_{2} + r_{3}) = 16 R^{2} \end{array}$

$\begin{array}{l} \Rightarrow r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} + 2 s^{2} - 2 (a b + b c + c a) + 2 s^{2} = 16 R^{2} \\ \Rightarrow r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} = 16 R^{2} - (a^{2} + b^{2} + c^{2}) \end{array}$