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A triangle's centroid is the point that maximizes the product of the directed distances of a point from the triangle's sidelines. [ 20 ] Let A B C {\displaystyle ABC} be a triangle, let G {\displaystyle G} be its centroid, and let D , E , F {\displaystyle D,E,F} be the midpoints of segments B C , C A , A B , {\displaystyle BC,CA,AB,} respectively.
Where the centroid coordinates are marked as zero, the coordinates are at the origin, and the equations to get those points are the lengths of the included axes divided by two, in order to reach the center which in these cases are the origin and thus zero.
By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of a, b, c. This process is known as cyclicity. [4] [5] Every triangle center function corresponds to a unique triangle center. This correspondence is not bijective. Different functions may define ...
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 coordinate geometry, the Section formula is a formula used to find the ratio in which a line segment is divided by a point internally or externally. [1] It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc. [2] [3] [4] [5]
In fact, given any point in cartesian coordinates, we can use this fact to determine where this point is with respect to a triangle. If a point lies in the interior of the triangle, all of the Barycentric coordinates lie in the open interval (,).
For a triangle the centroid can be easily computed, e.g. using cartesian coordinates. Weighting computes as simplex-to-cell area ratios. Three dimensions: The centroid of a tetrahedron is found as the intersection of three bisector planes and can be expressed as a matrix-vector product. Weighting computes as simplex-to-cell volume ratios.
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 ...