|
Miquel's theorem is a result in geometry, named after Auguste Miquel,〔A high school teacher in the French countryside (Nantua) according to 〕 concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal ''Journal de mathématiques pures et appliquées''. Formally, let ''ABC'' be a triangle, with arbitrary points ''A´'', ''B´'' and ''C´'' on sides ''BC'', ''AC'', and ''AB'' respectively (or their extensions). Draw three circumcircles to triangles ''AB´C´'', ''A´BC´'', and ''A´B´C''. Miquel's theorem states that these circles intersect in a single point ''M'', called the Miquel point. In addition, the three angles ''MA´B'', ''MB´C'' and ''MC´A'' (green in the diagram) are all equal, as are the three complementary angles ''MA´C'', ''MB´A'' and ''MC´B''.〔 - Wells refers to Miquel's theorem as the pivot theorem〕 The theorem (and its corollary) follow from the properties of two cyclic quadrilaterals drawn from any two of a triangle's vertices, having an edge in common as shown in the figure. Their combined angles at ''M'' (opposite ''A'' and opposite ''C'') will be (180 - ''A'') + (180 - ''C''), giving an exterior angle equal to (''A'' + ''C''). Since (''A'' + ''C'') also equals (180 - ''B''), the intersection at ''M'', lying on the chord ''A´C´'', must also lie on a cyclic quadrilateral passing through points ''B'', ''A´'', and ''C´''. This completes the proof. ==Pivot theorem== If in the statement of Miquel's theorem the points ''A´'', ''B´'' and ''C´'' form a triangle (that is, are not collinear) then the theorem was named the ''Pivot theorem'' in . (In the diagram these points are labeled ''P'', ''Q'' and ''R''.) If ''A´'', ''B´'' and ''C´'' are collinear then the Miquel point is on the circumcircle of ∆ABC and conversely, if the Miquel point is on this circumcircle, then ''A´'', ''B´'' and ''C´'' are on a line. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Miquel's theorem」の詳細全文を読む スポンサード リンク
|