【数学A】例題3.1.3：角の二等分線（One More）★★★

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% 例題A3.1.3：角の二等分線（One More）★★★
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$\triangle\mathrm{ABC}$の辺$\mathrm{BC}$を$\mathrm{AB}:\mathrm{AC}$に内分する点$\mathrm{P}$をとる．このとき，$\mathrm{AP}$は$\angle\mathrm{A}$の二等分線であることを示せ．

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% 例題A3.1.3の解答（One More）★★★
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$\triangle\mathrm{ABC}$において，辺$\mathrm{BA}$の延長上に$\mathrm{AC}=\mathrm{AD}$となるように点$\mathrm{D}$をとる．

$\mathrm{BP}:\mathrm{PC}=\mathrm{AB}:\mathrm{AC}$のとき，$\mathrm{BP}:\mathrm{PC}=\mathrm{BA}:\mathrm{AD}$であるから，

\[
\mathrm{AP}\parallel\mathrm{DC}
\]

したがって，
\[
\angle\mathrm{BAP}=\angle\mathrm{ADC},\angle\mathrm{PAC}=\angle\mathrm{ACD}
\]

また，$\mathrm{AC}=\mathrm{AD}$より，
\[
\angle\mathrm{ADC}=\angle\mathrm{ACD}
\]

ゆえに，$\angle\mathrm{BAP}=\angle\mathrm{PAC}$

よって，APは$\angle\mathrm{A}$の二等分線である．$\blacksquare$

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% 問題A3.1.3：角の二等分線（One More）★★★
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$\triangle\mathrm{ABC}$の2辺$\mathrm{AB}$，$\mathrm{AC}$上に$\mathrm{DE}\parallel\mathrm{BC}$となるような2点$\mathrm{D}$，$\mathrm{E}$をとり，辺$\mathrm{BC}$の中点を$\mathrm{M}$とする．このとき，$\mathrm{MD}$が$\angle\mathrm{AMB}$の二等分線であれば，$\mathrm{ME}$は$\angle\mathrm{AMC}$の二等分線であることを示せ．

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$\triangle\mathrm{MAB}$において，$\mathrm{MD}$は$\angle\mathrm{AMB}$の二等分線であるから，$\mathrm{AD}:\mathrm{DB}=\mathrm{MA}:\mathrm{MB}\cdots(\mathrm{i})$

$\triangle\mathrm{ABC}$において，$\mathrm{DE}\parallel\mathrm{BC}$であるから，

\[
\mathrm{AD}:\mathrm{DB}=\mathrm{AE}:\mathrm{EC}\cdots(\mathrm{ii})
\]

(i)，(ii)より，$\mathrm{AE}:\mathrm{EC}=\mathrm{MA}:\mathrm{MB}$

MはBCの中点であるから，$\mathrm{MB}=\mathrm{MC}$より，

\[
\mathrm{AE}:\mathrm{EC}=\mathrm{MA}:\mathrm{MC}
\]

よって，$\mathrm{ME}$は$\angle\mathrm{AMC}$の二等分線である．$\blacksquare$

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% 例題A3.1.3の側注：角の二等分線（One More）★★★
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% 側注1
平行線と線分の比の性質を利用する．

% 側注2
平行線における同位角，錯角はそれぞれ等しい．

% 側注3
$\triangle\mathrm{ACD}$は二等辺三角形である．

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% 問題A3.1.3の側注：角の二等分線（One More）★★★
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% 側注1
平行線と線分の比の定理を利用する．

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% 例題A3.1.3の考え方・補足など：角の二等分線（One More）★★★
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次のことを示せばよいと考えることができる．

$\mathrm{BP}:\mathrm{PC}=\mathrm{AB}:\mathrm{AC}\Longrightarrow\mathrm{AP}$は$\angle\mathrm{A}$の二等分線である．

線分の比に関する条件から，角が等しいことを示すことで証明する．このとき，辺$\mathrm{BA}$の点$\mathrm{A}$側の延長上に，$\mathrm{AC}=\mathrm{AD}$となるように点$\mathrm{D}$をとるなどして，平行線を上手く利用するとよい．

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線分の比から，角度が等しいことを証明する．