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http:///mymemo.xyz/wiki/api.php?action=feedcontributions&feedformat=atom&user=114.49.15.159 2024-05-28T12:02:11Z 利用者の投稿記録 MediaWiki 1.31.0 2017-07-15T17:17:56Z

<p>114.49.15.159: typo</p> <hr /> <div>{{出典の明記|date=2016年5月}}<br /> [[Image:Incircle and Excircles.svg|thumb|三角形（黒）&lt;br /&gt;[[内接円]]（青）と内心(I)&lt;br /&gt;傍接円（オレンジ）と傍心(J&lt;sub&gt;A&lt;/sub&gt;,J&lt;sub&gt;B&lt;/sub&gt;,J&lt;sub&gt;C&lt;/sub&gt;)&lt;br /&gt;内角の二等分線（赤）と外角の二等分線（緑）]]<br /> [[初等幾何学]]において&#039;&#039;&#039;三角形の内接円&#039;&#039;&#039;（さんかくけいのないせつえん、{{lang-en-short|&#039;&#039;triangle incircle&#039;&#039;}}）とは、その[[三角形]]の内部にあり3辺に接する[[円 (数学)|円]]である。三角形の内部にある円の中で最も面積が大きい円である。内接円の中心を&#039;&#039;&#039;内心&#039;&#039;&#039; (&#039;&#039;triangle incenter&#039;&#039;) と呼ぶ。<br /> <br /> &#039;&#039;&#039;傍接円&#039;&#039;&#039;（ぼうせつえん、{{lang-en-short|&#039;&#039;triangle excircle&#039;&#039;}}）は、三角形の外側にあり1辺と他の2辺の延長線に接する円である。傍接円の中心を&#039;&#039;&#039;傍心&#039;&#039;&#039; (&#039;&#039;triangle excenter&#039;&#039;) と呼ぶ。全ての三角形は、各辺に接する合計3つの傍接円を持つ。<br /> <br /> 内心は、3つの角の[[二等分線]]上にある。傍心は、1つの角の二等分線と他の2つの角の外角の2等分線上にある。内心と傍心は「三角形の3つの頂点と[[垂心]]」という位置関係にある。<br /> <br /> == 三角形の面積との関係 ==<br /> 内接円と傍接円の半径は、三角形の[[面積]]に関係している。<br /> <br /> &#039;&#039;S&#039;&#039; を三角形の面積、&#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039;, &#039;&#039;c&#039;&#039; を3辺の長さとしたとき、[[ヘロンの公式]]から、<br /> :&lt;math&gt;<br /> \begin{align}<br /> S &amp; {} = \frac{1}{4}\sqrt{(a+b+c)(a-b+c)(b-c+a)(c-a+b)} \\<br /> &amp; {} = \sqrt{s(s-a)(s-b)(s-c)}<br /> \end{align}<br /> &lt;/math&gt;<br /> ただし、&#039;&#039;s&#039;&#039; = (&#039;&#039;a&#039;&#039; + &#039;&#039;b&#039;&#039; + &#039;&#039;c&#039;&#039;)/2 とする。<br /> <br /> 内接円の半径は、<br /> : &lt;math&gt;\frac{2S}{a+b+c} = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}&lt;/math&gt;<br /> 辺 &#039;&#039;a&#039;&#039; に対する傍接円の半径は、<br /> : &lt;math&gt;\frac{2S}{c-a+b}&lt;/math&gt;<br /> となる。<br /> <br /> 他の2辺に対する傍接円の半径は同様に<br /> : &lt;math&gt;\frac{2S}{a-b+c}&lt;/math&gt;, &lt;math&gt;\frac{2S}{b-c+a}&lt;/math&gt;<br /> となる。<br /> <br /> これらの式から、傍接円は内接円より大きいことと最も長い辺に対応する傍接円が最も大きいことが分かる。<br /> <br /> == 内接円に関連する点 ==<br /> [[Image:Intouch Triangle and Gergonne Point.svg|thumb|△&#039;&#039;ABC&#039;&#039;と内接円（青）と内心（青 &#039;&#039;I&#039;&#039;）&lt;br /&gt;接点と接点から作られる三角形（赤 &#039;&#039;T&lt;sub&gt;a&lt;/sub&gt;&#039;&#039;, &#039;&#039;T&lt;sub&gt;b&lt;/sub&gt;&#039;&#039;, &#039;&#039;T&lt;sub&gt;c&lt;/sub&gt;&#039;&#039;）とジェルゴンヌ点（緑）]]<br /> <br /> === 九点円とフォイエルバッハ点 ===<br /> 内接円と傍接円は、[[九点円]]と接する。この接点を[[フォイエルバッハ点]]という。<br /> <br /> === ジェルゴンヌ点とジェルゴンヌ三角形 ===<br /> [[頂点]]を &#039;&#039;A&#039;&#039;, &#039;&#039;B&#039;&#039;, &#039;&#039;C&#039;&#039; とし、内接円が各辺と接する点を &#039;&#039;T&lt;sub&gt;A&lt;/sub&gt;&#039;&#039;, &#039;&#039;T&lt;sub&gt;B&lt;/sub&gt;&#039;&#039;, &#039;&#039;T&lt;sub&gt;C&lt;/sub&gt;&#039;&#039; とする（&#039;&#039;T&lt;sub&gt;A&lt;/sub&gt;&#039;&#039; は &#039;&#039;A&#039;&#039; の対辺上にあるとする）。⊿&#039;&#039;T&lt;sub&gt;A&lt;/sub&gt;T&lt;sub&gt;B&lt;/sub&gt;T&lt;sub&gt;C&lt;/sub&gt;&#039;&#039; をジェルゴンヌ三角形という。元の三角形の内接円はこの三角形の[[外接円]]になる。3直線 &#039;&#039;AT&lt;sub&gt;A&lt;/sub&gt;&#039;&#039;, &#039;&#039;BT&lt;sub&gt;B&lt;/sub&gt;&#039;&#039;, &#039;&#039;CT&lt;sub&gt;C&lt;/sub&gt;&#039;&#039; は1点で交わる。この点を[[ジェルゴンヌ点]]という。<br /> <br /> &#039;&#039;AT&lt;sub&gt;A&lt;/sub&gt;&#039;&#039;, &#039;&#039;BT&lt;sub&gt;B&lt;/sub&gt;&#039;&#039;, &#039;&#039;CT&lt;sub&gt;C&lt;/sub&gt;&#039;&#039; が1点で交わることから、⊿ABC が不等辺三角形のとき「AB と T&lt;sub&gt;A&lt;/sub&gt;T&lt;sub&gt;B&lt;/sub&gt; の交点」「CA と T&lt;sub&gt;C&lt;/sub&gt;T&lt;sub&gt;A&lt;/sub&gt; の交点」「BC と T&lt;sub&gt;B&lt;/sub&gt;T&lt;sub&gt;C&lt;/sub&gt; の交点」は同一直線上にある。この線をジェルゴンヌ線という。⊿ABC が二等辺三角形のときは上記の辺の組のうち1つが平行になるが、残りの2点を結ぶことで直線が定義できる。⊿ABC が正三角形の場合はこの直線は定義できない。<br /> <br /> {{clear}}<br /> <br /> == 内心の座標 ==<br /> [[直交座標系|座標平面]]における内心の座標は、3頂点の重み付き平均の値として求めることができる。<br /> <br /> 3頂点の座標を (x&lt;sub&gt;a&lt;/sub&gt;,y&lt;sub&gt;a&lt;/sub&gt;), (x&lt;sub&gt;b&lt;/sub&gt;,y&lt;sub&gt;b&lt;/sub&gt;), (x&lt;sub&gt;c&lt;/sub&gt;,y&lt;sub&gt;c&lt;/sub&gt;)、3辺の長さを a, b, c としたとき、<br /> :&lt;math&gt;\bigg(\frac{a x_a+b x_b+c x_c}{a+b+c},\frac{a y_a+b y_b+c y_c}{a+b+c}\bigg) = \frac{a}{a+b+c}(x_a,y_a)+\frac{b}{a+b+c}(x_b,y_b)+\frac{c}{a+b+c}(x_c,y_c)&lt;/math&gt;.<br /> となる。<br /> <br /> *[[三線座標]] であらわすと 1 : 1 : 1<br /> **絶対三線座標では r : r : r<br /> *[[重心座標]] であらわすと &#039;&#039;a&#039;&#039; : &#039;&#039;b&#039;&#039; : &#039;&#039;c&#039;&#039;<br /> となる。<br /> <br /> == 円の式 ==<br /> x : y : z を[[三線座標]]であらわしたときの点の座標は、u = cos&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;(A/2)&#039;&#039;, v = cos&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;(B/2)&#039;&#039;, w = cos&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;(C/2)&#039;&#039; とすると、円上の点に対して以下の式が成り立つ。<br /> * 内接円: &#039;&#039;u&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;x&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + v&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;y&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + w&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;z&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; - 2vwyz - 2wuzx - 2uvxy = 0&#039;&#039;<br /> :* A に対する傍接円: &#039;&#039;u&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;x&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + v&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;y&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + w&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;z&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; - 2vwyz + 2wuzx + 2uvxy = 0&#039;&#039;<br /> :* B に対する傍接円: &#039;&#039;u&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;x&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + v&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;y&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + w&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;z&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + 2vwyz - 2wuzx + 2uvxy = 0&#039;&#039;<br /> :* C に対する傍接円: &#039;&#039;u&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;x&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + v&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;y&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + w&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039;z&#039;&#039;&lt;sup&gt;&#039;&#039;2&#039;&#039;&lt;/sup&gt;&#039;&#039; + 2vwyz + 2wuzx - 2uvxy = 0&#039;&#039; <br /> <br /> == その他の関係 ==<br /> * 傍心同士・内心と傍心の中点は全て[[外接円]]上にある。<br /> * 3つの傍接円の半径の逆数の和は、内接円の半径の逆数に等しい（ルーリエの定理）。<br /> ** [[四面体]]と内接・傍接球、あるいはさらに高次の単体と内接・傍接球に対しても同様の関係が成り立つ。&lt;!--俗にアレクシの定理と呼ぶ{{要出典}}--&gt;<br /> <br /> == 関連項目 ==<br /> *[[三角形]]/[[三角形の中心]]<br /> *[[内接円]]<br /> *[[外接円]]<br /> *[[二等分線]]<br /> <br /> == 外部リンク ==<br /> * {{PlanetMath|urlname=ProofOfTriangleIncenter|title=proof of triangle incenter}}<br /> * {{ProofWiki|urlname=Definition:Incircle_of_Triangle|title=Definition:Incircle of Triangle}}<br /> * {{ProofWiki|urlname=Definition:Excircle_of_Triangle|title=Definition:Excircle of Triangle}}<br /> <br /> {{DEFAULTSORT:さんかくけいのないせつえんとほうせつえん}}<br /> [[Category:円 (数学)]]<br /> [[Category:三角形|ないせつえんとほうせつえん]]<br /> [[Category:初等幾何学]]<br /> [[Category:数学に関する記事]]</div>

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