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The only primitive Pythagorean triangles for which the square of the perimeter equals an integer multiple of the area are (3, 4, 5) with perimeter 12 and area 6 and with the ratio of perimeter squared to area being 24; (5, 12, 13) with perimeter 30 and area 30 and with the ratio of perimeter squared to area being 30; and (9, 40, 41) with ...
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
When working in polar coordinates it is not necessary to convert to Cartesian coordinates to use line integration, since the line integral between consecutive vertices (r i,θ i) and (r i+1,θ i+1) of a polygon is given directly by r i r i+1 sin(θ i+1 − θ i)/2. This is valid for all values of θ, with some decrease in numerical accuracy ...
Given a triangle with sides of length a, b, and c, if a 2 + b 2 = c 2, then the angle between sides a and b is a right angle. For any three positive real numbers a, b, and c such that a 2 + b 2 = c 2, there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.
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, given a triangle ABC, there exist unique points A´, B´, and C´ on the sides BC, CA, AB respectively, such that: [1] A´, B´, and C´ partition the perimeter of the triangle into three equal-length pieces. That is, C´B + BA´ = B´A + AC´ = A´C + CB´. The three lines AA´, BB´, and CC´ meet in a point, the trisected ...
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
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);