{%hackmd @RintarouTW/About %}

# Apollonius's Identity

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$$
已知
\cases{
\overline{AD} = a\\
\overline{BD} = b\\
\overline{AB} = c\\
\overline{PD} = d\\
\overline{BC} = e\\
\overline{CD} = f\\
\overline{AP} = \overline{PB} = \frac{1}{2}c
}
$$

$$
\begin{array}l
a^2+b^2&=(\overline{AC}^2+f^2) + (e^2+f^2)\\
&=((c+e)^2+f^2)+(e^2+f^2)\\
&=c^2+2ce+2e^2+2f^2\\
&=c^2+2(ce+e^2+f^2)\\
d^2&=\overline{PC}^2+f^2\\
&=(\frac{1}{2}c+e)^2+f^2\\
&=\frac{1}{4}c^2+(ce+e^2+f^2)\\
a^2+b^2-2d^2
&=c^2-\frac{1}{2}c^2\\
&=\frac{1}{2}c^2\\
a^2+b^2&=\frac{1}{2}c^2+2d^2
\end{array}
$$

###### tags: `math` `geometry` `apollonius`