加比の理
以下のような条件付き恒等式のことを，加比の理といいます．
定理．
\begin{aligned}
\frac{a}{b} = \frac{c}{d} = \frac{e}{f}
\end{aligned}
のとき，次の等式が成り立つ．
\begin{aligned}
\frac{a}{b} = \frac{ap + cq}{bp +dq} = \frac{ap + cq + er}{bp +dq +fr}
\end{aligned}
特に，p = q = r = 1のとき，
\begin{aligned}
\frac{a}{b} = \frac{a + c}{b +d} = \frac{a + c + e}{b +d +f}
\end{aligned}
証明．
\begin{aligned}
\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k
\end{aligned}
とおく．
a = bk, c =dk, e = fkとなるので，
\begin{aligned}
\frac{a}{b} = k
\end{aligned}
\begin{aligned}
\frac{ap + cq}{bp +dq} &= \frac{bkp + dkq}{bp +dq} \\
&= \frac{k(bp + dq)}{bp +dq} \\
&= k
\end{aligned}
\begin{aligned}
\frac{ap + cq + er}{bp +dq +fr} &= \frac{bkp + dkq + fkr}{bp + dq + fr} \\
&= \frac{k(bp + dq + fr)}{bp + dq + fr} \\
&= k
\end{aligned}
よって，
\begin{aligned}
\frac{a}{b} = \frac{ap + cq}{bp +dq} = \frac{ap + cq + er}{bp +dq +fr}
\end{aligned}
また，この等式にp = q = r = 1を代入すると，以下の等式が得られる．
\begin{aligned}
\frac{a}{b} = \frac{a + c}{b +d} = \frac{a + c + e}{b +d +f}
\end{aligned}
