Bisector

In planar geometry , the bisector (also called the symmetrical angle ) of an angle is the half-line that runs through the vertex of the angle and divides the angular field into two congruent parts.

An intersecting pair of straight lines defines two bisectors, in this case straight lines that are orthogonal to each other . Each of these bisectors is an axis of symmetry of the geometric figure, which is formed by the intersecting pair of lines. This symmetry property is followed by a characterization of the two bisectors as a geometric location , which is referred to as the bisector set .

In analytic geometry and in the analysis , the bisecting line of play coordinate axes of a Cartesian coordinate system a special role. The ones who are affected by the I. and III. Quadrant is called 1st bisector or 1st median , the other 2nd bisector .

In synthetic geometry , the bisectors of an intersecting pair of lines are also defined by their properties as axes of symmetry. The existence of this bisector is one of the axioms that characterize a freely movable pre-Euclidean plane .

Bisector in planar geometry

construction

An angle is given by its two legs (half-straight lines with a common beginning at the vertex of the angle). Then the bisector can be constructed with a pair of compasses and a ruler (set square ): A circle with any radius is drawn around the vertex . The compass is reattached at the intersection with the legs of the square. Then you draw a circle with the same radius. The intersections of these two circles lie on the bisector.

In this construction it is used that the bisector is at the same time the vertical line in the isosceles triangle that is given by the vertex and the first two auxiliary points.

If, more generally, there are two straight lines that intersect at a point, we have four angles and thus four bisectors. The bisectors of two vertex angles coincide; so only two bisectors remain. These two bisectors - which are orthogonal to each other - are called the bisectors of the two straight lines .

If we come back to the case of an angle that is bounded by two legs (half-straight lines) and now extend these legs to straight lines, then we get two straight lines with two bisectors. One of these is the bisector of the original angle; the other is the bisector of its minor angle ; it is called the outer angle bisector of the original angle.

Bisector theorem

The union of the two bisectors of an intersecting pair of straight lines is the set of all points that have the same distance from the two straight lines, or, to put it another way, the set of the centers of all circles that touch the two straight lines .

Bisector in the triangle

The 3 outer angles:
The 3 intersection points E, F, G lie on a straight line (red) and the following route conditions apply:

{\ displaystyle {\ begin {aligned} {\ frac {| EB |} {| EC |}} & = {\ frac {| AB |} {| AC |}} \\ {\ frac {| FB |} { | FA |}} & = {\ frac {| CB |} {| CA |}} \\ {\ frac {| DA |} {| DC |}} & = {\ frac {| BA |} {| BC |}} \ end {aligned}}}

If angle bisectors are mentioned in triangle theory, this term mostly refers to the interior angles , less often to the exterior angles . Here the bisector of an interior angle is often abbreviated as. This abbreviation also stands for the distance on the bisector that lies within the triangle and, in design tasks, for its length. ${\ displaystyle \ alpha}$ ${\ displaystyle w _ {\ alpha}}$

The following sentences apply to these bisectors :

The three bisectors (the interior angle) of a triangle intersect at one point. This point is the center of the inscribed circle (see also: Excellent points in a triangle ).
Each bisector (of an interior angle) in the triangle divides the opposite side in relation to the adjacent sides. (This statement is also known as the bisector theorem and can be proven with the help of similar triangles or by applying the sine law )
For the length of the bisector of an interior angle and for the adjacent sides of the length and , the relationship applies ${\ displaystyle w}$ ${\ displaystyle \ gamma}$ ${\ displaystyle a}$ ${\ displaystyle b}$
${\ displaystyle {\ frac {2 \ cos {\ frac {\ gamma} {2}}} {w}} = {\ frac {1} {a}} + {\ frac {1} {b}}}$ .
The bisectors of an interior angle and the exterior angle of a triangle belonging to the other two interior angles each intersect at a point. This point is the center of a circle .
The points of intersection of the outer angle bisector with the extended opposite sides of the corresponding inner angle lie, if they exist, on a straight line.

Bisector in a square

The bisecting a rectangle defining a generally quadrilateral . In the case of the tangent square it has degenerated into a point. In the case of the tendon quadrilateral, the enclosed quadrilateral is orthodiagonal . The bisector of a parallelogram generally includes a rectangle , the bisector of a rectangle a square , the bisector of an isosceles trapezoid a dragon square , the bisector of a quadrangle with equal opposite angles an isosceles trapezoid.

Angle bisector of a coordinate system

In a Cartesian coordinate system , the two bisectors of the coordinate axes play a special role:

The straight line with the equation is called the first bisector (bisector of the 1st and 3rd quadrant ) . This graph is the straight line through the origin with a slope of 1. It is called the 1st median in Austria . ${\ displaystyle y = x}$

The straight line with the equation is called the second bisector (bisector of the 2nd and 4th quadrant) . This graph is the straight line through the origin with the slope −1. ${\ displaystyle y = -x}$

Synthetic geometry

In synthetic geometry, a pre-Euclidean plane is an affine plane over a body whose characteristic is not 2, together with an orthogonality relation without isotropic lines between the lines of the plane. (Vertical) axis mirroring can be defined in such a plane (→ see Mirroring (geometry) #axis mirroring ). ${\ displaystyle K ^ {2}}$ ${\ displaystyle K}$ ${\ displaystyle \ perp}$

The following statement is called the bisector axiom :

For two straight , there is a straight line , so that when the mirror image of precisely the on the straight that is mapped. ${\ displaystyle a, b}$ ${\ displaystyle w}$ ${\ displaystyle w}$ ${\ displaystyle a}$ ${\ displaystyle b}$

If the straight lines are parallel and different, then their central parallel is a straight line which has the required symmetry property. Since central parallels always exist in a pre-Euclidean plane, the essential requirement is for an axis of symmetry for an intersecting pair of straight lines, i.e. for an angle bisector. From the existence of an angle bisector, the existence of exactly a second line always follows, which is perpendicular to the first. ${\ displaystyle a, b}$

A pre-Euclidean plane that fulfills the bisector axiom is called a floating plane .

literature

Friedrich Bachmann : Structure of geometry from the concept of reflection. 2nd edition, Berlin; Göttingen; Heidelberg 1973

Summary: To justify the geometry from the concept of reflection. Mathematische Annalen, Vol. 123, 1951, pp. 341ff

Wendelin Degen and Lothar Profke: Fundamentals of affine and Euclidean geometry , Teubner, Stuttgart, 1976, ISBN 3-519-02751-8

Individual evidence

↑ Degen (1976), p. 144

Web links

Commons : bisector - collection of images, videos and audio files

Wiktionary: bisector - explanations of meanings, word origins, synonyms, translations

Wiktionary: Angular symmetrals - explanations of meanings, word origins, synonyms, translations

Classic transversals - inside and outside angle bisector under D.7

[1] Degen (1976), p. 144