设旁切圆圆心为O,作OD⊥AC,交AC延长线于D,连接OA、OC,则：
OD=r,∠OCD=(π-C)/2,∠OAD=A/2,
CD=OD/tan∠OCD=r*tanC/2,
AD=OD/tan∠OAD=r*ctgA/2,
AC=AD-CD
=r(ctgA/2-tanC/2)
=[r*cos(A+C)/2]/[sin(A/2)*cos(C/2)]
=r*sin(B/2)/[sin(A/2)*cos(C/2)],
由正弦定理：
AC=b=2RsinB=r*sin(B/2)/[sin(A/2)*cos(C/2)],
——》r/R=[2sinB*sin(A/2)*cos(C/2)]/sin(B/2)
=4sin(A/2)*cos(B/2)*cos(C/2),
命题得证.