Câu a. Ta có: $S=pr=\frac{abc}{4R}$$\Rightarrow \frac{r}{R}=\frac{abc}{4pR^2}=\frac{8R^3sinA.sinB.sinC}{4R^3(sinA+sinB+sinC)}=\frac{2sinA.sinB.sinC}{sinA+sinB+sinC}$ $(*)$
Ta phải chứng minh: $sinA+sinB+sinC=2sin\frac{A+B}{2}.cos\frac{A-B}{2}+2sin\frac{C}{2}.cos\frac{C}{2}$
$=2cos\frac{C}{2}(cos\frac{A-B}{2}+sin\frac{C}{2})=2cos\frac{C}{2}(cos\frac{A-B}{2}+cos\frac{A+B}{2})$
$=4cos\frac{A}{2}.cos\frac{B}{2}.cos\frac{C}{2}$
Từ đó $(*)=\frac{r}{R}=\frac{16sin\frac{A}{2}.sin\frac{B}{2}.sin\frac{C}{2}.cos\frac{A}{2}.cos\frac{B}{2}.cos\frac{C}{2}}{4cos\frac{A}{2}.cos\frac{B}{2}.cos\frac{C}{2}}$
$\Rightarrow r=R.4sin\frac{A}{2}.sin\frac{B}{2}.sin\frac{C}{2}$ (đpcm)