Semiperimeter

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.

If A, B, B', C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (AA', BB', CC', shown in red in the diagram) are known as splitters, and The three splitters concur at the Nagel point of the triangle.

So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter.

The area A of any triangle is the product of its inradius (the radius of its inscribed circle) and its semiperimeter: The area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Heron's formula: The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths: This formula can be derived from the law of sines.

The four sides of a bicentric quadrilateral are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius.

In any triangle, the distance along the boundary of the triangle from a vertex to the point on the opposite edge touched by an excircle equals the semiperimeter.