$r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} =$

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

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

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

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

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Solution

The correct option is B $16 R^{2} - (a^{2} + b^{2} + c^{2})$
$(r_{1} + r_{2} + r_{3} - r) = 4 R$
$(r_{1} + r_{2} + r_{3} - r)^{2} = 16 R^{2}$
$r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} + 2 r_{1} r_{2} + 2 r_{2} r_{3} + 2 r_{3} r_{1} - 2 r [r_{1} + r_{2} + r_{3}] = 16 R^{2}$
$r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} + 2 (r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}) - 2 r [\frac{a b + b c + a c - s^{2}}{r}] = 16 R^{2}$
$r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + 2 s^{2} - 2 [a b + b c + a c - s^{2}] = 16 R^{2}$
$r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + 4 s^{2} - 2 (a b + b c + a c) = 16 R^{2}$
$r_{1}^{2} + r_{2}^{2} + r_{3}^{2} = 16 R^{2} - 4 s^{2} + 2 (a b + b c + a c)$
$= 16 R^{2} - (a + b + c)^{2} + 2 a b + 2 b c + 2 a c$
$= 16 R^{2} - (a^{2} + b^{2} + c^{2})$

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