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In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s.
where = (+ + +) is the semiperimeter. Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
where (h, k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x, y).The semi-major axis is the mean value of the maximum and minimum distances and of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis
where s, the semiperimeter, is defined to be = + + +. This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a ...
A splitter of a triangle is a cevian (a segment from a vertex to the opposite side) that divides the perimeter into two equal lengths, this common length being called the semiperimeter of the triangle. The three splitters of a triangle all intersect each other at the Nagel point of the triangle.
where r is the inradius, and s is the semiperimeter (in fact, this formula holds for all tangential polygons), and [15]: Lemma 2 = = = where ,, are the radii of the excircles tangent to sides a, b, c respectively. We also have
The semiperimeter s of a Heronian triangle cannot be prime (as () () is the square of the area, and the area is an integer, if s were prime, it would divide another factor; this is impossible as these factors are all less than s).
A triangle with sides <, semiperimeter = (+ +), area, altitude opposite the longest side, circumradius, inradius, exradii,, tangent to ,, respectively, and medians,, is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.