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辺と面の関係

立体図形では、辺と面は、3つの関係になれる。交わるか、平行か、含まれるかだ。

例えば、直方体ABCD-EFGHで、辺と面の関係を、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}

辺と面が交わる

辺ABが交わるのは、面辺ADHEと面BCGHだ。

交わるとは、1点を共有する、ともいえる。

\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 @{.}(0,24)}, \end{xy}

<例題 \( \Large 1 \) >直方体ABCD-EFGHで、辺BF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}

<解答 \( \Large 1 \) >
    面ABCD    面EFGH

辺と面が平行

辺ABが平行なのは、面辺CDHGと面EFGHだ。

平行な辺と平行な面は、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 @{-}(0,-18)}, {(0,24) \ar @{-}(20,12)}, {(-20,12) \ar @{-}(-20,-6)}, {(-20,-6) \ar @{-}(0,-18)}, {(-20,-6) \ar @{.}(0,6)}, {(0,6) \ar @{.}(20,-6)}, {(0,-18) \ar @{-}(20,-6)}, \end{xy}

<例題 \( \Large 2 \) >直方体ABCD-EFGHで、辺BF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}

<解答 \( \Large 2 \) >
    面ADHE    面CDHG

辺が面に含まれる

辺ABが含まれるのは、面辺ABCDと面ABFEだ。

\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", (0,0)*{\bullet}="A", (20,12)*{\bullet}="A", (-20,12)*{\bullet}="A", (20,-6)*{\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)}, {(0,6) \ar @{.}(20,-6)}, {(0,6) \ar @{.}(0,24)}, \end{xy}

<例題 \( \Large 3 \) >直方体ABCD-EFGHで、辺BFが含まれる面を、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}

<解答 \( \Large 3 \) >
    面ABFE    面BCGF

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