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The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object X {\displaystyle X} in n {\displaystyle n} - dimensional space is the intersection of all hyperplanes that divide X {\displaystyle X} into two parts of equal moment about the hyperplane.
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.
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.
By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral. The area of the Steiner ellipse equals the area of the triangle times 4 π 3 3 , {\displaystyle {\frac {4\pi }{3{\sqrt {3}}}},} and hence is 4 times the ...
A filled triangular area with a base width of b, height h and top vertex displacement a, with respect to an axis through the centroid: The figure presents a triangle with dimensions 'b', 'h' and 'a', along with axes 'x' and 'y' that pass through the centroid.
The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). The first eight centered triangular numbers on a hex grid
Let the centroids of these triangles be X, Y, Z respectively. Then the lines AX, BY, CZ are concurrent. The point of concurrence N 1 is the first Napoleon point, or the outer Napoleon point, of the triangle ABC. The triangle XYZ is called the outer Napoleon triangle of ABC. Napoleon's theorem asserts that this triangle is an equilateral triangle.
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.