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In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to meet opposite sides at D, E, F respectively. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments,
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 ...
The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: [19] Any side of a triangle is greater than or equal to the difference between the other two sides. In the case of a normed vector space, the statement is:
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,
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.
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} .
In geometry, the Euler line, named after Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər), is a line determined from any triangle that is not equilateral.It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]
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