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Concept# Semiperimeter

Summary

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.
Motivation: triangles
The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, c
:s = \frac{a+b+c}{2}.
Properties
In any triangle, any vertex and the point where the opposite excircle touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has a length equal to the semiperimeter. If A, B, B', C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency ({{overline|AA'}}, {{overline|BB'}}, {{overline

Official source

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