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面と面の関係
立体図形では、面と面は、2つの関係になれる。交わるか、平行か、だ。
例えば、直方体ABCD-EFGHで、面と面の関係を、考えていこう。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
面と面が交わる
面ABFEが交わるのは、4つの面だ。
面ABFEと、面ABCDと面EFGH
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
さらに、面ABFEと、面ADHEと面BCGF
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
面が交わるとは、直線を共有する、ともいえる。
なお、交わる面のうち、垂直に交わる面もある。
<例題 \( \Large 1 \) >直方体ABCD-EFGHで、面ABCDと交わる面を、すべて書きなさい。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
面と面が平行
面ABCDが平行なのは、面CFGHだ。
平行な面同士は、どこまで延長しても、交わらない。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
\end{xy}
<例題 \( \Large 2 \) >直方体ABCD-EFGHで、面ABFEと平行な面を、すべて書きなさい。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
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