r

r_{a}

r_{b}

r_{c}

ABC

BC=a

AC=b

AB=c

S

ABC

\frac{1}{r}=\frac{1}{r_{a}}+\frac{1}{r_{b}}+\frac{1}{r_{c}}

S=\sqrt{rr_{a}r_{b}r_{c}}

S=pr

p

\frac{1}{r}=\frac{p}{S}

O

r_{a}

BC=a

ABC

Q

AB=c

AC=b

M

N

S=S_{\triangle AOB}+S_{\triangle AOC}-S_{\triangle BOC}=\frac{1}{2}AB\cdot OM+\frac{1}{2}AC\cdot ON-\frac{1}{2}BC\cdot OQ=

=\frac{1}{2}c\cdot r_{a}+\frac{1}{2}b\cdot r_{a}-\frac{1}{2}a\cdot r_{a}=\frac{c+b-a}{2}r_{a}=(p-a)r_{a}.

r_{a}=\frac{S}{p-a}

r_{b}=\frac{S}{p-b}

r_{c}=\frac{S}{p-c}

\frac{1}{r_{a}}+\frac{1}{r_{b}}+\frac{1}{r_{c}}=\frac{p-a}{S}+\frac{p-b}{S}+\frac{p-c}{S}=\frac{p-a+p-b+p-c}{S}=\frac{3p-2p}{S}=\frac{p}{S}=\frac{1}{r}.

S=\sqrt{p(p-a)(p-b)(p-c)}=\sqrt{\frac{S}{r}\cdot\frac{S}{r_{a}}\cdot\frac{S}{r_{b}}\cdot\frac{S}{r_{c}}}=\sqrt{\frac{S^{4}}{rr_{a}r_{b}r_{c}}}=\frac{S^{2}}{\sqrt{rr_{a}r_{b}r_{c}}}.

\sqrt{rr_{a}r_{b}r_{c}}=S