三角形内两两相切的圆

王守恩

记：3条边为 \(a,b,c,\ \) 3个角为 \( 2A,2B,2(A+B),\ \) 3个圆半径为 \( r_{1},r_{2},r_{3},\)

则有：\(\frac{a}{\sin(2A)}=\frac{b}{\sin(2B)}=\frac{c}{\sin(2A+2B)}\)

\(c=2\sqrt{r_{1}\ r_{2}\ }+r_{1}\cot(A)+r_{2}\cot(B)\)

\(a=2\sqrt{r_{2}\ r_{3}\ }+r_{2}\cot(B)+r_{3}\tan(A+B)\)

\(b=2\sqrt{r_{3}\ r_{1}\ }+r_{3}\tan(A+B)+r_{1}\cot(A)\)

或：3条边为 \(\sin(2A),\sin(2B),\sin(2A+2B),\ \)3个圆半径为 \( r_{1},r_{2},r_{3},\)

\(\sin(2A+2B)=2\sqrt{r_{1}\ r_{2}\ }+r_{1}\cot(A)+r_{2}\cot(B)\)

\(\sin(2A)=2\sqrt{r_{2}\ r_{3}\ }+r_{2}\cot(B)+r_{3}\tan(A+B)\)

\(\sin(2B)=2\sqrt{r_{3}\ r_{1}\ }+r_{3}\tan(A+B)+r_{1}\cot(A)\)

或：