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In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle. Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.〔.〕〔(Encyclopedia of Triangle Centers ), accessed 2014-10-28.〕 For polygons with more than three sides, the incenter only exists for tangential polygons—those that have an incircle that is tangent to each side of the polygon. In this case the incenter is the center of this circle and is equally distant from all sides. ==Definition and construction== It is a theorem in Euclidean geometry that the three interior angle bisectors of a triangle meet in a single point. In Euclid's ''Elements'', Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle. The incircle itself may be constructed by dropping a perpendicular from the incenter to one of the sides of the triangle and drawing a circle with that segment as its radius.〔Euclid's ''Elements, (Book IV, Proposition 4: To inscribe a circle in a given triangle ). David Joyce, Clark University, retrieved 2014-10-28.〕 The incenter lies at equal distances from the three line segments forming the sides of the triangle, and also from the three lines containing those segments. It is the only point equally distant from the line segments, but there are three more points equally distant from the lines, the excenters, which form the centers of the excircles of the given triangle. The incenter and excenters together form an orthocentric system.〔.〕 The medial axis of a polygon is the set of points whose nearest neighbor on the polygon is not unique: these points are equidistant from two or more sides of the polygon. One method for computing medial axes is using the grassfire transform, in which one forms a continuous sequence of offset curves, each at some fixed distance from the polygon; the medial axis is traced out by the vertices of these curves. In the case of a triangle, the medial axis consists of three segments of the angle bisectors, connecting the vertices of the triangle to the incenter, which is the unique point on the innermost offset curve.〔.〕 The straight skeleton, defined in a similar way from a different type of offset curve, coincides with the medial axis for convex polygons and so also has its junction at the incenter.〔.〕 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「Incenter」の詳細全文を読む スポンサード リンク
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