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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.
Subdivide a triangle ABC arbitrarily into a triangulation consisting of smaller triangles meeting edge to edge. Then a Sperner coloring of the triangulation is defined as an assignment of three colors to the vertices of the triangulation such that Each of the three vertices A, B, and C of the initial triangle has a distinct color
A conjecture of Fiorini and Wilson that every triangle-free planar graph, other than the claw K 1,3, is not uniquely 3-edge-colorable. A 2012 conjecture that if G is a d-regular planar multigraph, then G is d-edge-colorable if and only if G is oddly d-edge-connected. This conjecture is a generalization of the four color theorem, which arises at ...
The theorem cannot be generalized to all nonplanar triangle-free graphs: not every nonplanar triangle-free graph is 3-colorable. In particular, the Grötzsch graph and the Chvátal graph are triangle-free graphs requiring four colors, and the Mycielskian is a transformation of graphs that can be used to construct triangle-free graphs that ...
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 ...
Marcella Nasseri found her missing brother Thomas Manizak through a USA TODAY story after 25 years. New details emerge of his background.
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