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The triangle medians and the centroid.. In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. . Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's cent
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. Ross Honsberger called its existence "one of the crown jewels of modern geometry". [1] In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth ...
In the diagram, the medians (in black) intersect at the centroid G. Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, L . This point is called the triangle's symmedian point , or alternatively the Lemoine point or Grebe point .
The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). [6] The centroid divides each of the medians in the ratio 2 : 1 , {\displaystyle 2:1,} which is to say it is located 1 3 {\displaystyle {\tfrac {1}{3}}} of the distance from each side to the opposite ...
In a cyclic quadrilateral, four line segments, each perpendicular to one side and passing through the opposite side's midpoint, are concurrent. [3]: p.131, [5] These line segments are called the maltitudes, [6] which is an abbreviation for midpoint altitude. Their common point is called the anticenter.
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or does not exist (if the lines are parallel). Other types ...
This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. [23] The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. [24]
The intersection point of both midlines will be the centroid of the tetrahedron. Since a tetrahedron has six edges in three opposite pairs, one obtains the following corollary: [8] In a tetrahedron, the three midlines corresponding to opposite edge midpoints are concurrent, and their intersection point is the centroid of the tetrahedron.