$(r_{1} - r) (r_{2} - r) (r_{3} - r) = 4 r^{2} R$

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Solution

$(r_{1} - r) (r_{2} - r) (r_{3} - r) = (r_{1} r_{2} - r r_{1} - r r_{2} + r^{2}) (r_{3} - r) = r_{1} r_{2} r_{3} - r r_{1} r_{3} - r r_{2} r_{3} + r^{2} r_{3} - r r_{1} r_{2} + r^{2} r_{1} + r^{2} r_{2} - r^{3} = s^{2} r - r (r_{1} r_{3} + r_{2} r_{3} + r_{1} r_{3}) + r^{2} (r_{1} + r_{2} + r_{3}) - r^{3} = s^{2} r - r (s^{2}) + r^{2} (4 R + r) - r^{2} = s^{2} r - s^{2} r + 4 R r^{2} + r^{3} - r^{3} = 4 r^{2} R$

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(r_{1} - r) (r_{2} - r) (r_{3} - r) = 4 R r^{2}

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