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The top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y. In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the ...
A polygon is a generalization of a 3-sided triangle, a 4-sided quadrilateral, and so on to n sides. A hypercube is a generalization of a 2-dimensional square, a 3-dimensional cube, and so on to n dimensions. A quadric, such as a hypersphere, ellipsoid, paraboloid, or hyperboloid, is a generalization of a conic section to higher dimensions.
Ordinary trigonometry studies triangles in the Euclidean plane .There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions [broken anchor], definitions via differential equations [broken anchor], and definitions using functional equations.
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment,
Generalization for arbitrary triangles, green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure ...
In May 2021, Dao Thanh Oai gave a generalization of the isogonal conjugate as follows: [2] Let ABC be a triangle, P a point on its plane and Ω an arbitrary circumconic of ABC. Lines AP, BP, CP cut again Ω at A', B', C' respectively, and parallel lines through these points to BC, CA, AB cut Ω again at A", B", C" respectively.
A generalization of the Sierpiński triangle can also be generated using Pascal's triangle if a different modulus is used. Iteration n {\displaystyle n} can be generated by taking a Pascal's triangle with P n {\displaystyle P^{n}} rows and coloring numbers by their value modulo P {\displaystyle P} .
The Jacobi point is a generalization of the Fermat point, which is obtained by letting α = β = γ = 60° and ABC having no angle being greater or equal to 120°. If the three angles above are equal, then N lies on the rectangular hyperbola given in areal coordinates by
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