Knowra Hadwiger–Finsler inequality Hadwiger–Finsler inequality For a triangle with side lengths a, b, c and area K, the Hadwiger–Finsler inequality states that a² + b² + c² ≥ 4√3 K. Equality holds for an equilateral triangle.
Heron's formula : A formula expressing a triangle’s area from its three side lengths: K = √(s(s−a)(s−b)(s−c)), where s is the semiperimeter. It converts the inequality’s area term into an expression involving only the three side lengths.
Triangle : A polygon with three sides and three interior angles. The inequality applies specifically to these plane figures and their side lengths.
Euler's inequality : The triangle inequality R ≥ 2r, relating a triangle’s circumradius R to its inradius r. It is another classical bound on triangle geometry, but compares radii rather than area and side lengths.
Triangle inequality problem : A problem asking whether three given positive lengths can be the sides of a triangle. The inequality can provide an additional necessary check on candidate side lengths and area.
Law of cosines : A relation connecting a triangle’s side lengths to the cosine of one of its angles. Substituting its side-length identity exposes how angle geometry constrains the inequality.
Triangle area : The measure of the two-dimensional region enclosed by a triangle. Area is the quantity controlled by the sum of squared side lengths.
Pedoe's inequality : A geometric inequality relating the areas and circumradii of two triangles. It offers a different area comparison, involving a pair of triangles instead of one.
Maximum triangle area : The largest area achievable under specified constraints on a triangle’s dimensions. For a fixed sum of squared sides, the inequality gives a sharp upper bound, attained by an equilateral triangle.
Equilateral triangle : A triangle whose three sides and three angles are equal. It is the equality case, attaining the smallest possible ratio of squared side sum to area.
Semiperimeter : Half the perimeter of a polygon; for a triangle with sides a, b, c, it is (a+b+c)/2. It appears in Heron’s formula, a standard route for proving the inequality.
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