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For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio. [1] [2] The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.
[3]: p.233, Lemma 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 medial triangle. [4]: p.139 The medial triangle is the only inscribed triangle for which none of the other three interior triangles has smaller area. [5]: p. 137
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 ...
In geometry, the midpoint polygon of a polygon P is the polygon whose vertices are the midpoints of the edges of P. [1] [2] It is sometimes called the Kasner polygon after Edward Kasner, who termed it the inscribed polygon "for brevity". [3] [4] The medial triangle The Varignon parallelogram
[3] [5] A little earlier than Kepler, Pedro Nunes wrote about it in 1567, and it is "likely to have been widespread in late medieval and Renaissance manuscript traditions". [3] It has also been independently rediscovered several times, later than Kepler. [1] A right triangle formed by an edge midpoint, base center point, and apex of a square ...
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
Hence [3]: p.126 + = (+). This is also a corollary to the parallelogram law applied in the Varignon parallelogram. The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas.