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Question Number 188320 by normans last updated on 27/Feb/23

i f t h e o n △ A B C,

i n n e r c i r c l e r a d i u s a n d o u t e r c i r c l e r a d i u s r a n d R

s h o w t h a t;

r = 4 R s i n \frac{A}{2} \cdot s i n \frac{B}{2} \cdot s i n \frac{C}{2}

Answered by mr W last updated on 28/Feb/23

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R

Δ = \frac{a b c}{4 R} = 2 R^{2} \sin A \sin B \sin C

r = \frac{2 Δ}{a + b + c}

= \frac{2 Δ}{2 R (\sin A + \sin B + \sin C)}

= \frac{2 R^{2} \sin A \sin B \sin C}{R (\sin A + \sin B + \sin C)}

= \frac{2 R \sin A \sin B \sin C}{\sin A + \sin B + \sin C}

= \frac{2 R \sin A \sin B \sin C}{4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}

= \frac{16 R \sin \frac{A}{2} \cos \frac{A}{2} \sin \frac{B}{2} \cos \frac{B}{2} \sin \frac{C}{2} \cos \frac{C}{2}}{4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}

= 4 R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}

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