# Bisectrix

2020 Mathematics Subject Classification: *Primary:* 51M15 [MSN][ZBL]

*of an angle; angle bisector*

The half-line (ray) issuing from the apex of the angle and bisecting it. In other words, the bisectrix of an angle is the set of points located inside the angle and equally distant from both of its sides. A bisectrix of a triangle is the segment (and also its length) of the bisectrix of an internal angle of the triangle from the apex to the point of intersection with the opposite side. A bisectrix of a triangle divides a side of the triangle into segments that are proportional to the adjacent sides. The bisectrices of a triangle intersect at one point, which is the *incentre*: the centre of the inscribed circle in the triangle. The quadruple of points $A,B,K,L$, consisting of two apices $A,B$ of the triangle $ABC$, the point $K$ of intersection of the bisectrix of the angle $C$ with $AB$, and the point $L$ of intersection of the bisectrix of the external angle $C$ with $AB$ forms a harmonic quadruple of points.

#### Comment

This line is also known as the *(angle) bisector*. The bisector of the external of supplementary angle is the *external bisector*: it lies at a right angle to the internal bisector.

#### References

- Robert Bix, "Topics in Geometry" Elsevier (2016)
**ISBN**1483296466 - H. S. M. Coxeter, Samuel L. Greitzer, "Geometry Revisited" New Mathematical Library
**19**Mathematical Association of America (1967)**ISBN**0883856190 Zbl 0166.16402

**How to Cite This Entry:**

Bisectrix.

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