ケイシーの定理
cos  ∠ K i O K j = 1 − 2 sin 2  ∠ K i O K j 2 = 1 − 2 ⋅ ( K i K j ̄ 2 R ) 2 = 1 − K i K j ̄ 2 2 R 2 {\displaystyle \cos \angle K_{i}OK_{j}=1-2\sin ^{2}{\frac {\angle K_{i}OK_{j}}{2}}=1-2\cdot \left({\frac {\overline {K_{i}K_{j}}}{2R}}\right)^{2}=1-{\frac {{\overline {K_{i}K_{j}}}^{2}}{2R^{2}}}}
以上を余弦定理の式に代入すると
O i O j ̄ 2 = ( R − R i ) 2 + ( R − R j ) 2 − 2 ( R − R i ) ( R − R j ) ( 1 − K i K j ̄ 2 2 R 2 ) = ( R − R i ) 2 + ( R − R j ) 2 − 2 ( R − R i ) ( R − R j ) + ( R − R i ) ( R − R j ) ⋅ K i K j ̄ 2 R 2 = ( ( R − R i ) − ( R − R j ) ) 2 + ( R − R i ) ( R − R j ) ⋅ K i K j ̄ 2 R 2 {\displaystyle {\begin{aligned}{\overline {O_{i}O_{j}}}^{2}&=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})\left(1-{\frac {{\overline {K_{i}K_{j}}}^{2}}{2R^{2}}}\right)\\&=(R-R_{i})^{2}+(R-R_{j})^{2}-2(R-R_{i})(R-R_{j})+(R-R_{i})(R-R_{j})\cdot {\frac {{\overline {K_{i}K_{j}}}^{2}}{R^{2}}}\\&=((R-R_{i})-(R-R_{j}))^{2}+(R-R_{i})(R-R_{j})\cdot {\frac {{\overline {K_{i}K_{j}}}^{2}}{R^{2}}}\end{aligned}}}
よって
t i j = O i O j ̄ 2 − ( R i − R j ) 2 = R − R i ⋅ R − R j ⋅ K i K j ̄ R {\displaystyle t_{ij}={\sqrt {{\overline {O_{i}O_{j}}}^{2}-(R_{i}-R_{j})^{2}}}={\frac {{\sqrt {R-R_{i}}}\cdot {\sqrt {R-R_{j}}}\cdot {\overline {K_{i}K_{j}}}}{R}}}
が得られる。円に内接する四角形 K 1 K 2 K 3 K 4 {\displaystyle \,K_{1}K_{2}K_{3}K_{4}} にトレミーの定理を用いて変形すると