アポロニウスの円
⇔ n 2 | P A → | 2 = m 2 | P B → | 2 . {\displaystyle \Leftrightarrow n^{2}|{\overrightarrow {\mathrm {PA} }}|^{2}=m^{2}|{\overrightarrow {\mathrm {PB} }}|^{2}.}
⇔ ( n P A → + m P B → ) ⋅ ( n P A → − m P B → ) = 0. {\displaystyle \Leftrightarrow (n{\overrightarrow {\mathrm {PA} }}+m{\overrightarrow {\mathrm {PB} }})\cdot (n{\overrightarrow {\mathrm {PA} }}-m{\overrightarrow {\mathrm {PB} }})=0.}
⇔ n P A → + m P B → n + m ⋅ n P A → − m P B → n − m = 0. {\displaystyle \Leftrightarrow {\frac {n{\overrightarrow {\mathrm {PA} }}+m{\overrightarrow {\mathrm {PB} }}}{n+m}}\cdot {\frac {n{\overrightarrow {\mathrm {PA} }}-m{\overrightarrow {\mathrm {PB} }}}{n-m}}=0.