こんにちは。相城です。今回は三角形の内角の二等分線の比について書いておきます。
内角の二等分線の比の公式
内角の二等分線の比の公式
例題をやってみよう
実際問題をやってみよう。
下の図の△ABCにおいて内心をIとし, AIの延長線と辺BCとの交点をDとする。このとき, BDの長さと, AI : IDの比を求めなさい。
【解説】
内心Iは三角形の内角の二等分線の交点なので, 直線AIは
![Rendered by QuickLaTeX.com \angle A](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-627c1f66d526e118634c9727d08d5fa8_l3.png)
したがって,
AB : AC
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
AB : AC
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
BD : DC
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
よって,
BD
![Rendered by QuickLaTeX.com =\dfrac{8}{8+5}\times7=\dfrac{56}{13}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-af60aa105b6f45315a5e1bc8336248e3_l3.png)
BIもまた,
![Rendered by QuickLaTeX.com \angle{B}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b630288b6eb549b07321dcbed4862035_l3.png)
AI : ID
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
AI : ID
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com \dfrac{56}{13}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f1759ebe2796ac07dae4b3039a247417_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
よって, BD
![Rendered by QuickLaTeX.com =\dfrac{56}{13}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-31bf3fb74a06b2612bacbfcfe286ad7d_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
公式にしてみよう
公式化してみよう
AI : IDを公式化してみよう。Iは△ABCの内心, DはAIの延長線とBCとの交点
BD : DC
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com c](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4b0923aa4784625c1c704c5edfbd409a_l3.png)
![Rendered by QuickLaTeX.com b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7d3b74b93cf8d3562b537c81c243a48c_l3.png)
なので, BD
![Rendered by QuickLaTeX.com =a\times\dfrac{c}{b+c}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b2980a039b2da4cf9878e04aaf006954_l3.png)
BD
![Rendered by QuickLaTeX.com =\dfrac{ac}{b+c}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8887574540b1278867e7129cebe312f3_l3.png)
AI : ID
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
AI : ID
![Rendered by QuickLaTeX.com = c : \dfrac{ac}{b+c}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-df0150d211d5daa5e3ab09fa3729a57e_l3.png)
![Rendered by QuickLaTeX.com b+c](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-73e8664b52f0546e42d6e93a9aff938e_l3.png)
![Rendered by QuickLaTeX.com c](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4b0923aa4784625c1c704c5edfbd409a_l3.png)
AI : ID
![Rendered by QuickLaTeX.com = (b+c) : a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3d09054c31edc52651bcfdcc86af1991_l3.png)
という公式が得られます。
実際上の例題で,
AI : ID
![Rendered by QuickLaTeX.com = (8+5) : 7 = 13 : 7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b5ffbb9d600a5f0302788676ac61d301_l3.png)
となります。
AI : IDの公式
△ABC(BC
, CA
, AB
)の内心をIとし, 線分AIの延長線と辺BCとの交点をDとするとき,
AI : ID
![Rendered by QuickLaTeX.com =a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ef5e023296fbcd93fca7ae7d8fb99e99_l3.png)
![Rendered by QuickLaTeX.com =b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-81fba85cf686691bdff648859f638669_l3.png)
![Rendered by QuickLaTeX.com =c](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9dbab1b5e2afc23adec9c82b4edb6b32_l3.png)
AI : ID
![Rendered by QuickLaTeX.com = (b+c) : a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3d09054c31edc52651bcfdcc86af1991_l3.png)