For many problems of elementary geometry in which it comes to angles on circles, the below stated terms and statements can be used.
By connecting the end points A and B different from each other a circular arc with its center point M and the point P on the arc, the following angles are:
- Circumferential angle or peripheral angle ( Φ ) is called an angle whose vertex P lies on the one arc that complements the given arc via [ DOWN] to complete circle ( the circumcircle of triangle ABP).
- Central angle ( μ ): If M is the midpoint of the given arc, so the angle is referred to as the corresponding central angle ( central angle ).
- A tendon tangent angle ( τ ) at the given arc is bounded by the tendon [AB ] and the circle tangent at point A or B.
Many authors of geometry textbooks do not take at peripheral angles, central angles and tendon tangent angles with respect to a given arc, but on a given chord [AB ]. If one such definition used, they must distinguish two types of circumferential angles, namely acute and obtuse peripheral angle. Is defined as a central angle, in this case the smaller of the two angles which are included by the radii [mA] and [ MB]. The wording of the sentences in the next section must be vary slightly when using this definition.
Circle angle set ( Zentriwinkelsatz )
The center angle ( center angle ) of a circular arc is twice the size of one of the associated circumferential angle ( circumferential angle ).
The proof of this statement is particularly simple in the right outlined special case. The two angles at B and P are as base angles in the isosceles triangle MBP equal. The third angle of the triangle MBP ( with the vertex M) has the size. The sentence about the angle sum is consequently and on about how claims.
M in the general case not situated on a limb of the circumferential angle. The line then divides PM circumferential angle and central angle into two angles ( and or and ), each individually the statement applies to the, since the conditions of the proved special case are met. Therefore, the statement also applies to the entire circumference and the entire angle of central angle. In addition, the validity of the peripheral angle theorem allows (see below) a transfer of the general case in the special case, without limiting the generality of the services already provided for the special case of the proof.
A very important special case is when the given circular arc is a semi-circle: In this case, the central angle is 180 ° ( a straight angle ), while the circumferential angle equal to 90 °, that is a right angle. Thus, the theorem of Thales proves a special case of the circle angle set.
Circumferential angle set ( peripheral angle theorem )
All circumferential angle ( peripheral angle) over a circular arc are equal. This arc is then called barrel arc.
The peripheral angle theorem is a direct consequence of the circle angle theorem: Every circumferential angle is half as large as the central angle ( central angle ) to the circle angle set. So all circumferential angle must be equal.
However, it may be necessary to prove the peripheral angle theorem in another way, otherwise it is not used as a condition in the proof of the circle angle set.
Tendons tangent angle set
The two chord tangent angle of a circular arc as large as the corresponding angle at circumference ( circumferential angle ), and half as large as the corresponding angle at the center ( center angle ).
Proof see Web Links
Use in construction tasks
In particular, the circumferential angle set can be often used for geometrical constructions. In many cases, examined the amount ( the locus ) of all points P, from which a given distance (here, [AB] ) appears at a particular angle. The desired point set generally consists of two arcs of a circle, the so-called barrel arcs.