【数学A】例題3.2.12：2つの円の共通接線の作図（One More）★★★

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% 例題A3.2.12：2つの円の共通接線の作図（One More）★★★
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右の図のように，半径がそれぞれ$r,r^{\prime}(r>r^{\prime})$である2つの円$\mathrm{O},\mathrm{O}^{\prime}$がある．この$2$つの円の共通外接線を作図せよ．

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% 例題A3.2.12の解答（One More）★★★
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[1]線分$\mathrm{OO}^{\prime}$を直径とする円をかく．

[2]$\mathrm{O}$を中心とする半径$r-r^{\prime}$の円をかく．

[3][1]の円と[2]の円の交点を$\mathrm{P},\mathrm{Q}$とする．

[4]半直線$\mathrm{OP},\mathrm{OQ}$と円$\mathrm{O}$の交点を，それぞれ$\mathrm{A},\mathrm{C}$とする．また，点$\mathrm{O}^{\prime}$を通り，線分$\mathrm{OA},\mathrm{OC}$に平行な直線と円$\mathrm{O}^{\prime}$との交点を，それぞれ$\mathrm{B},\mathrm{D}$とする．このとき，直線$\mathrm{AB}$と直線$\mathrm{CD}$を引くと，この2直線が2つの円$\mathrm{O},\mathrm{O}^{\prime}$の共通外接線である．

$\angle\mathrm{OPO}^{\prime}=90^{\circ},\mathrm{AP}=\mathrm{OA}-\mathrm{OP}=r-(r-r^{\prime})=r^{\prime}$であり，$\mathrm{OA}\parallel\mathrm{O}^{\prime}\mathrm{B}$であるから，四角形$\mathrm{APO}^{\prime}\mathrm{B}$は長方形となる．

したがって，$\angle\mathrm{OAB}=\angle\mathrm{O}^{\prime}\mathrm{BA}=90^{\circ}$

よって，直線$\mathrm{AB}$は2つの円$\mathrm{O},\mathrm{O}^{\prime}$の共通外接線である．

直線$\mathrm{CD}$についても同様に示される．

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% 問題A3.2.12：2つの円の共通接線の作図（One More）★★★
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右の図のように，半径がそれぞれ$r,r^{\prime}(r>r^{\prime})$である2つの円$\mathrm{O},\mathrm{O}^{\prime}$がある．この$2$つの円の共通内接線を作図せよ．

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% 問題A3.2.12の解答（One More）★★★
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[1]線分$\mathrm{OO}^{\prime}$を直径とする円をかく．

[2]$\mathrm{O}$を中心とする半径$r+r^{\prime}$の円をかく．

[3][1]の円と[2]の円の交点を$\mathrm{P},\mathrm{Q}$とする．

[4]半直線$\mathrm{OP},\mathrm{OQ}$と円$\mathrm{O}$の交点を，それぞれ$\mathrm{A},\mathrm{C}$とする．また，点$\mathrm{O}^{\prime}$を通り，線分$\mathrm{OA},\mathrm{OC}$に平行な直線と円$\mathrm{O}^{\prime}$との交点を，それぞれ$\mathrm{B},\mathrm{D}$とする．このとき，直線$\mathrm{AB}$と直線$\mathrm{CD}$を引くと，この2直線が2つの円$\mathrm{O},\mathrm{O}^{\prime}$の共通内接線である．

$\angle\mathrm{OPO}^{\prime}=90^{\circ},\mathrm{AP}=\mathrm{OP}-\mathrm{OA}=(r+r^{\prime})-r=r^{\prime}$であり，$\mathrm{OA}\parallel\mathrm{O}^{\prime}\mathrm{B}$であるから，四角形$\mathrm{APO}^{\prime}\mathrm{B}$は長方形となる．

したがって，$\angle\mathrm{OAB}=\angle\mathrm{O}^{\prime}\mathrm{BA}=90^{\circ}$

よって，直線$\mathrm{AB}$は2つの円$\mathrm{O},\mathrm{O}^{\prime}$の共通内接線である．

直線$\mathrm{CD}$についても同様に示される．

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% 例題A3.2.12の側注：2つの円の共通接線の作図（One More）★★★
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% 側注1
垂直二等分線を引く．

% 側注2
点$\mathrm{X}$を中心として半径$r,r^{\prime}$の2円をかき，点$\mathrm{X}$から引いた半直線と各円との交点を結ぶと，長さ$r-r^{\prime}$の線分が得られる．

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% 問題A3.2.12の側注：2つの円の共通接線の作図（One More）★★★
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% 側注1
垂直二等分線を引く．

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% 例題A3.2.12の考え方・補足など：2つの円の共通接線の作図（One More）★★★
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% 考え方
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2つの円が，共通接線$l$の同じ側にあるとき，この$l$を2つの円の共通外接線という．一方，2つの円が共通接線$l$の両側にあるとき，この$l$を2つの円の共通内接線という．なお，ここでは2つの円は離れているから，共通外接線は2本あることに注意すること．