# Periphery

in even geometry is a periphery a circle, which goes through all corner points of a polygon (a Polygons).

Not for each Polygon such a periphery exists. Generally a convex Polygon possesses exactly a periphery if itself the Mittelsenkrechten of all sides in onePoint cut. In this case the common point is the center of the Umkreises.

Each triangle possesses a periphery (see below). For squares, pentagons etc. does not apply this generally any longer. Squares, which have a periphery, are called chord squares. Special cases are give-lie to trapezoid, the rectangle and the square.

Independently of the corner number each regular polygon has a periphery. For the periphery radius of a regular [itex] n< /math> - Ecks with the side length [itex] A< /math> applies:

[itex] R = \ frac {A} {2 \ sin \ frac {180^ \ circ} {n}}< /math>

## periphery of a triangle

the center of the Umkreises in a triangle is the intersection of all three Mittelsenkrechten of the triangle, the circumcenter in such a way specified of the triangle.

The fact that for any triangle a periphery exists can be justified as follows: All points of the Mittelsenkrechtento [OFF] are equivalent far distant from A and B. Accordingly the points of the Mittelsenkrechten have agreeing distances of B and C to [UC]. The intersection of these two Mittelsenkrechten is thus from all three corners (A, B and C) equivalent far. It must thusalso on the third Mittelsenkrechten lie. If one draws a circle, which goes through a corner of the triangle around this intersection, then also the other corners must be on this circle.

For pointed-angular triangles the circumcenter lies inside the triangle. With the right-angled triangle is thatCenter of the hypotenuse at the same time circumcenter (see set of the Thales). In case of an blunt-angular triangle (with an angle over 90°) the circumcenter is outside of the triangle.

Circumcenter of a triangle ([itex] X_3< /math>)
trilinear coordinates [itex] \ cos \ alpha \: \, \ cos \ beta \: \, \ cos \ gamma< /math>

[itex] = A (b^2+c^2-a^2) \,: \, b (c^2+a^2-b^2) \: \, C (a^2+b^2-c^2)< /math>

Baryzentri coordinates [itex] \ sin (2 \ alpha) \: \, \ sin (2 \ beta) \: \, \ sin (2 \ gamma)< /math>

The periphery radius of a triangle can be computed with the following formulas:

[itex] R = \ frac {A} {2 \ sin \ alpha} = \ frac {b} {2 \ sin \ beta} = \ frac {C} {2 \ sin \ gamma}< /math>
[itex] R = \ frac {ABC} {4A}< /math>

The designations stand [itex] for A< /math>, [itex] b< /math>, [itex] C< /math>for the side lengths and [itex] \ alpha< /math>, [itex] \ beta< /math>, [itex] \ gamma< /math> for the sizes of the internal angles. [itex] A< /math> designates the area of the triangle, which can be computed with the help of the heronischen formula.

The circumcenter is like the emphasis and the elevator intersection on the Euler straight line.

## linguisticMeans “in the periphery”

in the geographical sense is spoken frequently of a periphery. Meant then the radius of a circle is around one point, not the diameter of the circle. Example: “Were evacuated the area in the periphery of 10 km around the reactor Tschernobyl.” The circle meant with ithas a radius of 10 kilometers and a diameter of 20 kilometers.

## related terms

the periphery is apart from the Inkreis and the three excircles the most well-known under the special circles of triangle geometry.

One transfers the definition of the Umkreises to (three-dimensional) , Then one receives the term of the Umkugel , thus a ball , to area on which all corner points of a given polyhedron (Vielflächners) lie.