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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.
The area of the triangle is times the length of any side times the perpendicular distance from the side to the centroid. [15] A triangle's centroid lies on its Euler line between its orthocenter and its circumcenter, exactly twice as close to the latter as to the former: [16] [17]
For an equilateral triangle, these are the same point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex. A strict definition of a triangle centre is a point whose trilinear coordinates are f ( a , b , c ) : f ( b , c , a ) : f ( c , a , b ) where f is a function of ...
In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates .
The best known and simplest formula is = /, where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. Euclid proved that the area of a triangle is ...
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 ...
Each median divides the area of the triangle in half, hence the name, and hence a triangular object of uniform density would balance on any median. (Any other lines that divide triangle's area into two equal parts do not pass through the centroid.) [2] [3] The three medians divide the triangle into six smaller triangles of equal area.
Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction.The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the ...