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The points of concurrence are called the Fermat-Torricelli points, sometimes denoted F 1 and F 2. The intersection of the Fermat line (i.e., that line joining the two Fermat-Torricelli points) and the Napoleon line (i.e., that line joining the two Napoleon points) is the triangle's symmedian point (ETC's X(6)).
An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median and a line segment joining the ...
It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name. The length of the shorter edges is 3 / 5 that of the longer edges. [2]
Consider a line segment AB as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires ...
A straight line in the plane of triangle ABC whose equation in trilinear coordinates has the form f ( a , b , c ) x + g ( a , b , c ) y + h ( a , b , c ) z = 0 where the point with trilinear coordinates ( f ( a , b , c ) : g ( a , b , c ) : h ( a , b , c ) ) is a triangle center, is a central line in the plane of triangle ABC relative to the ...
For instance, the Sylvester–Gallai theorem, stating that any non-collinear set of points in the plane has an ordinary line containing exactly two points, transforms under projective duality to the statement that any projective arrangement of finitely many lines with more than one vertex has an ordinary point, a vertex where only two lines cross.
Since barycentric coordinates are all positive for a point in a triangle's interior but at least one is negative for a point in the exterior, and two of the barycentric coordinates are zero for a vertex point, the barycentric coordinates given for the orthocenter show that the orthocenter is in an acute triangle's interior, on the right-angled ...
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.