Simplify $r^{2} + r_{1}^{2} + r_{2}^{2} + r_{3}^{2} = 16 R^{2} - a^{2} - b^{2} - c^{2}$

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Solution

$(r_{1} + r_{2} + r_{3} - r)^{2} = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} - 2 r (r_{1} + r_{2} + r_{3}) + 2 (r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1})$ ....(1)
We know that
$r_{1} + r_{2} + r_{3} - r = 4 R . . . . (2)$
and $(r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}) = s^{2} . . . . (3)$
$r (r_{1} + r_{2} + r_{3}) = \frac{Δ}{s} (\frac{Δ}{s - a} + \frac{Δ}{s - b} + \frac{Δ}{s - c})$
$= (\frac{Δ^{2}}{s (s - a)} + \frac{Δ^{2}}{s (s - b)} + \frac{Δ^{2}}{s (s - c)})$
$= (s - b) (s - c) + (s - a) (s - c) + (s - a) (s - b)$
$(∵ Δ = \sqrt{s (s - a) (s - b) (s - c)})$
$= 3 s^{2} - 2 (a + b + c) s + (a b + b c + c a)$
$= 3 s^{2} - 2 s (2 s) + (a b + b c + c a)$
$r (r_{1} + r_{2} + r_{3}) = - s^{2} + (a b + b c + c a) . . . (4)$
from $(1), (2), (3)$ and $(4)$
$(4 R)^{2} = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} - 2 (- s^{2} + (a b + b c + c a)) + 2 s^{2}$
$\Rightarrow r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} = 16 R^{2} - [4 s^{2} - 2 (a b + b c + c a)]$
$= 16 R^{2} - [(2 s)^{2} - (2 a b + 2 b c + 2 c a)]$
$= 16 R^{2} - [(a + b + c)^{2} - 2 a b - 2 b c - 2 a c]$
$= 16 R^{2} - [(a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a) - 2 a b - 2 b c - 2 c a]$
$= 16 R^{2} - a^{2} - b^{2} - c^{2}$

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