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A Heronian triangle is commonly defined as one with integer sides whose area is also an integer. The lengths of the sides of such a triangle form a Heronian triple (a, b, c) for a ≤ b ≤ c. Every Pythagorean triple is a Heronian triple, because at least one of the legs a, b must be even in a Pythagorean triple, so the area ab/2 is an integer.
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]
Moreover, (x, y, z) is a primitive Pythagorean triple if both of the following conditions are verified: [14] if x is odd then d is a square and { d , x 2 / d } are coprime odd integers, and if x is even then d /2 is a square and { d /2, x 2 /2 d } are coprime integers of different parities.
The fire triangle. The fire triangle or combustion triangle is a simple model for understanding the necessary ingredients for most fires. [1] The triangle illustrates the three elements a fire needs to ignite: heat, fuel, and an oxidizing agent (usually oxygen). [2] A fire naturally occurs when the elements are present and combined in the right ...
For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
Whereas Roberts's theorem concerns the fewest possible triangles made by a given number of lines, the related Kobon triangle problem concerns the largest number possible. [3] The two problems differ already for n = 5 {\displaystyle n=5} , where Roberts's theorem guarantees that three triangles will exist, but the solution to the Kobon triangle ...
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.
In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.