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In an acute triangle, the sum of the circumradius R and the inradius r is less than half the sum of the shortest sides a and b: [4]: p.105, #2690 + < +, while the reverse inequality holds for an obtuse triangle. For an acute triangle with medians m a, m b, and m c and circumradius R, we have [4]: p.26, #954
An equilateral triangle may have integer sides with three rational angles as measured in degrees, [13] known for the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes), [14] and the only triangle whose Steiner inellipse is a circle (specifically, the incircle).
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
Triangle; Automedian triangle; Delaunay triangulation; Equilateral triangle; Golden triangle; Hyperbolic triangle (non-Euclidean geometry) Isosceles triangle; Kepler triangle; Reuleaux triangle; Right triangle; Sierpinski triangle (fractal geometry) Special right triangles; Spiral of Theodorus; Thomson cubic; Triangular bipyramid; Triangular ...
The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle. The orthocenter of a triangle, usually denoted by H, is the point where the three (possibly extended) altitudes intersect. [1] [2] The orthocenter lies inside the triangle if and only if the triangle is acute.
An equilateral triangle base and three equal isosceles triangle sides It gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group C 3v , isomorphic to the symmetric group , S 3 .
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
In geometry, an isosceles triangle (/ aɪ ˈ s ɒ s ə l iː z /) is a triangle that has two sides of equal length or two angles of equal measure. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.