1r1 - 1r2 1r2 - 1r3 1r3 - 1r1 - 64R3a2 b2 c2

L.H.S. = (s a) + (s b)S . (s b) + (s a)S . (s c) + (s a)S = abcS^3, ∴ 2s = (a + b + c) etc. ...

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Relation between Inradius and Perimeter of Triangle

1r1 + 1r2 1r...

Question

$(\frac{1}{r_{1}} + \frac{1}{r_{2}}) (\frac{1}{r_{2}} + \frac{1}{r_{3}}) (\frac{1}{r_{3}} + \frac{1}{r_{1}}) = \frac{64 R^{3}}{a^{2} b^{2} c^{2}}$

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Solution

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Δ A B C (\frac{1}{r_{1}} + \frac{1}{r_{2}}) (\frac{1}{r_{2}} + \frac{1}{r_{3}}) (\frac{1}{r_{3}} + \frac{1}{r_{1}}) = \frac{K R^{3}}{a^{2} b^{2} c^{2}}

K

△ A B C,

(\frac{1}{r_{1}} + \frac{1}{r_{2}}) (\frac{1}{r_{2}} + \frac{1}{r_{3}}) (\frac{1}{r_{3}} + \frac{1}{r_{1}}) = \frac{K R^{3}}{a^{2} b^{2} c^{2}},

K

1 + r + r^{2} + r^{3} + . . . . . . . + r^{n} = (1 + r) (1 + r^{2}) (1 + r^{4}) (1 + r^{8})

△ A B C

(1 - \frac{r_{1}}{r_{2}}) (1 - \frac{r_{1}}{r_{3}}) = 2

△ A B C

r_{2}

r_{3}

r_{3}

r_{1}

r_{1}

r_{2}

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