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The Nagel point is the isotomic conjugate of the Gergonne point.The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line.The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.
In geometry, given a triangle ABC, there exist unique points A´, B´, and C´ on the sides BC, CA, AB respectively, such that: [1] A´, B´, and C´ partition the perimeter of the triangle into three equal-length pieces. That is, C´B + BA´ = B´A + AC´ = A´C + CB´. The three lines AA´, BB´, and CC´ meet in a point, the trisected ...
In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a', b', c'), or equivalently in ratio form, ka' : kb' : kc' for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0.
Some examples of the use of areal coordinates in triangle geometry, Mathematical Gazette 83, November 1999, 472–477. Schindler, Max; Chen, Evan (July 13, 2012). Barycentric Coordinates in Olympiad Geometry (PDF). Retrieved 14 January 2016. Clark Kimberling's Encyclopedia of Triangles Encyclopedia of Triangle Centers. Archived from the ...
The three splitters concur at the Nagel point of the triangle. A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. [15] The incircle radius is no greater than one-ninth the sum of the altitudes. [16]: 289
The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the triangle as stated ...
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.