リュイリエの定理

リュイリエの定理（英: L'Huilier's theorem) とは、初等幾何学における三角形についての定理で、1809年に^[1]スイスの数学者サイモン・アントワーヌ・ジャン・リュイリエによって提唱されたものである。

定理

リュイリエの定理 ― 三角形の内接円の半径を $r$ 、3つの傍接円の半径をそれぞれ $r_{\text{A}},r_{\text{B}},r_{\text{C}}$ とすると、

{\frac {1}{r}}={\frac {1}{r_{\text{A}}}}+{\frac {1}{r_{\text{B}}}}+{\frac {1}{r_{\text{C}}}}

証明

面積 $S$ の三角形の3辺を $a, b, c$ とする。

内接円の半径 $r$ の逆数は

{\frac {1}{r}}={\frac {a+b+c}{2S}}

3傍接円の半径 $r A, r B, r C$ の逆数は

{\frac {1}{r_{\text{A}}}}={\frac {-a+b+c}{2S}}

{\frac {1}{r_{\text{B}}}}={\frac {a-b+c}{2S}}

{\frac {1}{r_{\text{C}}}}={\frac {a+b-c}{2S}}

故に逆数和は

{\begin{aligned}{\frac {1}{r_{\text{A}}}}+{\frac {1}{r_{\text{B}}}}+{\frac {1}{r_{\text{C}}}}&={\frac {-a+b+c}{2S}}+{\frac {a-b+c}{2S}}+{\frac {a+b-c}{2S}}\\&={\frac {a+b+c}{2S}}={\frac {1}{r}}\end{aligned}}

となる。

派生項目

リュイリエは、彼の著書 (Lhuilier, 1809) において

S={\sqrt {r\cdot r_{\text{A}}\cdot r_{\text{B}}\cdot r_{\text{C}}{\vphantom {A}}}}

であることも示唆している。

これより

r_{\text{A}}\,r_{\text{B}}\,r_{\text{C}}={\frac {S^{2}}{r}}

であるから、リュイリエの定理：

{\frac {1}{r_{\text{A}}}}+{\frac {1}{r_{\text{B}}}}+{\frac {1}{r_{\text{C}}}}={\frac {1}{r}}

と辺々掛け合わせると

r_{\text{B}}\,r_{\text{C}}+r_{\text{C}}\,r_{\text{A}}+r_{\text{A}}\,r_{\text{B}}=s^{2}

が得られる。ここで $s$ は $△ABC$ の半周長 $(a + b + c)/2$ である。この等式は、カール・フォイエルバッハが1822年に得たものである^[1]^[2]。

脚注

^ ^a ^b 岩田至康『幾何学大辞典』 1巻、槙書店、1971年、15,193頁。
^ それよりも前にリュイリエが彼の著書 (Lhuilier, 1809) において全く同等の等式を示唆している（224頁）。

出典

Simon Lhuilier (1809). Elémens d'analyse géométrique et d'analyse algébrique, appliquées à la recherche des lieux géométriques. A Paris: chez J. J. Paschoud; à Genève: chez le même libraire. pp. 223-224. doi:10.3931/e-rara-4330

外部リンク

『図形の美しい３つの定理〜逆数の和〜』 - 高校数学の美しい物語

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