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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.
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 triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. [48] Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. [49]
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1 ⁄ 4 turn or 90 degrees).
The area (A) of a regular heptagon of side length a is given by: A = 7 4 a 2 cot π 7 ≃ 3.634 a 2 . {\displaystyle A={\frac {7}{4}}a^{2}\cot {\frac {\pi }{7}}\simeq 3.634a^{2}.} This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then ...
Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined. [13]: pp. 201–203
The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle. [6]: p.153 Figure 3. A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle. Figure 4
where ∆(APB) is the area of triangle APB. Denote the segments that the diagonal intersection P divides diagonal AC into as AP = p 1 and PC = p 2, and similarly P divides diagonal BD into segments BP = q 1 and PD = q 2. Then the quadrilateral is tangential if and only if any one of the following equalities are true: [30]