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In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of each side of the triangle * The foot of each altitude * The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes). The nine-point circle is also known as Feuerbach's circle, Euler's circle, Terquem's circle, the six-points circle, the twelve-points circle, the ''n''-point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle.〔 Kocik and Solecki (sharers of a 2010 Lester R. Ford Award) give a proof of the Nine-Point Circle Theorem.〕 == Significant nine points == File:Nine-point circle.svg The diagram above shows the nine significant points of the nine-point circle. Points ''D'', ''E'', and ''F'' are the midpoints of the three sides of the triangle. Points ''G'', ''H'', and ''I'' are the feet of the altitudes of the triangle. Points ''J'', ''K'', and ''L'' are the midpoints of the line segments between each altitude's vertex intersection (points ''A'', ''B'', and ''C'') and the triangle's orthocenter (point ''S''). For an acute triangle, six of the points (the midpoints and altitude feet) lie on the triangle itself; for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle. 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「Nine-point circle」の詳細全文を読む スポンサード リンク
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