In triangle , if $r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} = k R^{2} - (a^{2} + b^{2} + c^{2})$

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Detailed Solution

$\begin{array}{l} r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r^{2} = {(r_{1} + r_{2})}^{2} - 2 r_{1} r_{2} + {(r_{3} - r)}^{2} + 2 r_{3} r \\ = {(r_{1} + r_{2})}^{2} + {(r_{3} - r)}^{2} - 2 r_{1} r_{2} + 2 r_{3} r \\ = {(r_{1} + r_{2} + r_{3} - r)}^{2} - 2 (r_{1} + r_{2}) (r_{3} - r) - 2 r_{1} r_{2} + 2 r_{3} r \\ = (4 R)^{2} - 2 (\frac{Δ}{s - a} + \frac{Δ}{s - b}) (\frac{Δ}{s - c} - \frac{Δ}{s}) \\ - 2 \frac{Δ}{s - a} \frac{Δ}{s - b} + 2 \frac{Δ}{s - c} \frac{Δ}{s} \\ = 16 R^{2} - 2 Δ^{2} (\frac{s - b + s - a}{(s - a) (s - b)}) (\frac{s - s + c}{s (s - c)}) \\ - 2 s (s - c) + 2 (s - a) (s - b) \\ = 16 R^{2} - 2 Δ^{2} \frac{(c) (c)}{Δ^{2}} - 2 s^{2} + 2 s c + 2 s^{2} \end{array}$

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