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  2. Ceva's theorem - Wikipedia

    en.wikipedia.org/wiki/Ceva's_theorem

    [3] [4] Two proofs are given in the following. The first one is very elementary, using only basic properties of triangle areas. [3] However, several cases have to be considered, depending on the position of the point O. The second proof uses barycentric coordinates and vectors, but is somehow [vague] more natural and not case dependent.

  3. Pythagorean theorem - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_theorem

    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 ...

  4. Triangle inequality - Wikipedia

    en.wikipedia.org/wiki/Triangle_inequality

    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:

  5. Generalization - Wikipedia

    en.wikipedia.org/wiki/Generalization

    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.

  6. Napoleon's theorem - Wikipedia

    en.wikipedia.org/wiki/Napoleon's_theorem

    Napoleon's theorem: If the triangles centered on L, M, N are equilateral, then so is the green triangle.. In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle.

  7. Triangulated category - Wikipedia

    en.wikipedia.org/wiki/Triangulated_category

    A triangulated category is an additive category D with a translation functor and a class of triangles, called exact triangles [2] (or distinguished triangles), satisfying the following properties (TR 1), (TR 2), (TR 3) and (TR 4). (These axioms are not entirely independent, since (TR 3) can be derived from the others.

  8. 10 Classic Southern Holiday Recipes To Make Right Now

    www.aol.com/10-classic-southern-holiday-recipes...

    2. Hoppin’ John. Southerners are usually eating Hoppin’ John (a simmery mix of black-eyed peas and rice) on New Year's Day. Like most “vegetable” recipes from around this area, it contains ...

  9. Equivalence class - Wikipedia

    en.wikipedia.org/wiki/Equivalence_class

    Congruence is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.

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