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  2. Triangle inequality - Wikipedia

    en.wikipedia.org/wiki/Triangle_inequality

    Euclid's construction for proof of the triangle inequality for plane geometry. Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. [6] Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB.

  3. List of triangle inequalities - Wikipedia

    en.wikipedia.org/wiki/List_of_triangle_inequalities

    The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);

  4. Erdős–Mordell inequality - Wikipedia

    en.wikipedia.org/wiki/Erdős–Mordell_inequality

    Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007). Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from P to the sides are replaced by the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross ...

  5. Lebesgue's lemma - Wikipedia

    en.wikipedia.org/wiki/Lebesgue's_lemma

    The proof is a one-line application of the triangle inequality: ... where the last inequality uses the fact that u = Pu together with the definition of the operator norm

  6. Euler's theorem in geometry - Wikipedia

    en.wikipedia.org/wiki/Euler's_theorem_in_geometry

    Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed circle is reached for the equilateral triangle and only for it, is valid in absolute geometry. [7]

  7. Hadwiger–Finsler inequality - Wikipedia

    en.wikipedia.org/wiki/Hadwiger–Finsler_inequality

    Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF) [1] Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. a = b = c.

  8. Barrow's inequality - Wikipedia

    en.wikipedia.org/wiki/Barrow's_inequality

    Barrow's proof of this inequality was published in 1937, as his solution to a problem posed in the American Mathematical Monthly of proving the Erdős–Mordell inequality. [1] This result was named "Barrow's inequality" as early as 1961. [4] A simpler proof was later given by Louis J. Mordell. [5]

  9. Minkowski inequality - Wikipedia

    en.wikipedia.org/wiki/Minkowski_inequality

    1 Proof. 2 Minkowski's integral inequality. 3 Reverse inequality. 4 Generalizations to other functions. ... The Minkowski inequality is the triangle inequality in (). ...