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(In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares.) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle. [2]: p. 115 All triangles in which the Euler line is parallel to one side are acute. [3]
Triangle – 3 sides Acute triangle; Equilateral triangle; Heptagonal triangle; Isosceles triangle. Golden Triangle; Obtuse triangle; Rational triangle; Heronian triangle. Pythagorean triangle; Isosceles heronian triangle; Primitive Heronian triangle; Right triangle. 30-60-90 triangle; Isosceles right triangle; Kepler triangle; Scalene triangle ...
For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle. Fat objects are especially important in computational geometry.
The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. [1] It is an isosceles triangle which is obtuse with an irrational but algebraic ratio between the lengths of its sides and its base.
Every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with each other, so there are only two distinct inscribed squares. An obtuse triangle has a single inscribed square, with one side coinciding with part of the triangle's longest side.
For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure : not itself an element of a polytope, but a diagram showing how the elements meet.
If a 2 + b 2 < c 2, then the triangle is obtuse. Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language: sgn(α + β − γ) = sgn(a 2 + b 2 − c 2), where α is the angle opposite to side a, β is the angle opposite to side b, γ is the angle opposite to side c, and sgn is the sign function. [30]
For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an obtuse angle), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side, exterior ...