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In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).
Net. In four-dimensional geometry, the 24-cell is the convex regular 4-polytope [1] (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C 24, or the icositetrachoron, [2] octaplex (short for "octahedral complex"), icosatetrahedroid, [3] octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.
The trilinear coordinates for a point in the triangle is the ratio of all the distances to the triangle sides. Because the incenter is the same distance from all sides of the triangle, the trilinear coordinates for the incenter are [ 6 ] 1 : 1 : 1. {\displaystyle \ 1:1:1.}
A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides. [3]: p.139 For a given point inside that medial triangle, the inellipse with its center at that point is unique. [3]: p.142
The 8-vertex 16-cell has 4 long diameters inclined at 90° = 𝜋 / 2 to each other, often taken as the 4 orthogonal axes or basis of the coordinate system. The 24-vertex 24-cell has 12 long diameters inclined at 60° = 𝜋 / 3 to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 ...
One more interesting line (in some sense dual to the Newton's one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.
If a point lies on an edge of the triangle but not at a vertex, one of the area coordinates (the one associated with the opposite vertex) is zero, while the other two lie in the open interval (,). If the point lies on a vertex, the coordinate associated with that vertex equals 1 and the others equal zero.