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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle.. In geometry, an altitude of a triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite the apex (the base).
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the ...
A transversal produces 8 angles, as shown in the graph at the above left: 4 with each of the two lines, namely α, β, γ and δ and then α 1, β 1, γ 1 and δ 1; and; 4 of which are interior (between the two lines), namely α, β, γ 1 and δ 1 and 4 of which are exterior, namely α 1, β 1, γ and δ.
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). [1] Ptolemy used the theorem as an aid to creating his table of chords ...
Adjacent colored angles are equal in measure. The point N is the Jacobi point for triangle ABC and these angles. In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ABC and a triple of angles α, β, γ.
Kőnig's theorem is equivalent to many other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem. [14]
According to VSEPR theory, the geometry of a molecule can be predicted by counting how many electron pairs and atoms are connected to a central atom. [ 11 ] [ 12 ] Bent's rule states "[A]tomic s character concentrates in orbitals directed toward electropositive substituents". [ 2 ]