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A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. [ 1 ] [ 2 ] Heronian triangles are named after Heron of Alexandria , based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84 .
The same area formula can also be derived from Heron's formula for the area of a triangle from its three sides. However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between the semiperimeter and side length in those triangles. [20]
Various methods may be used in practice, depending on what is known about the triangle. Other frequently used formulas for the area of a triangle use trigonometry, side lengths (Heron's formula), vectors, coordinates, line integrals, Pick's theorem, or other properties. [3]
However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is , which is larger than three times , the area of a 26–14–14 isosceles triangle (all by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.
Examples of such proofs can be found in the articles Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem, Law of cosines, and others that are linked to here.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
Isosceles trapezoid; Triangle. Acute and obtuse triangles; Equilateral triangle; Euler's line; Heron's formula; Integer triangle. Heronian triangle; Isosceles triangle; List of triangle inequalities; List of triangle topics; Pedal triangle; Pedoe's inequality; Pythagorean theorem; Pythagorean triangle; Right triangle; Triangle inequality ...