Angle bisector of a triangle

What is the angle bisector of a triangle? On this question some people from the tongue breaks the notorious saying: "It's a rat running in the corners and dividing the angle in half." If the answer should be "with humor", then, perhaps, it is correct. But scientifically, the answer to this question would be something like this: "This is a ray starting at the top of the corner and dividing the latter into two equal parts." In geometry, this figure is also perceived as a segment of the bisectrix before its intersection with the opposite side of the triangle. This is not an erroneous opinion. And what else is known about the bisector of the angle, other than its definition?

As with any locus of points, it has its own characteristics. The first of these is, rather, not even a sign, but a theorem that can be summarized as follows: "If the bisectrix divides the opposite side into two parts, then their ratio will correspond to the ratio of the sides of the large triangle."

The second property that it has: the intersection point of the bisectors of all angles is called the center.

The third sign: bisectors of one inner and two outer corners of a triangle intersect at the center of one of the three inscribed circles in it.

The fourth property of the bisector of the angle of the triangle is that if each of them is equal, then the latter is isosceles.

The fifth sign also concerns an isosceles triangle and is the main reference point for its recognition in the drawing by bisectors, namely: in an isosceles triangle it simultaneously acts as a median and height.

The angle bisector can be constructed using a compass and a ruler:

The sixth rule says that it is impossible to build a triangle with the help of the latter only with the existing bisectors, as it is impossible to construct in this way a doubling of the cube, quadrature of the circle, and trisection of the angle. Strictly speaking, this is all the properties of the bisector of the angle of the triangle.

If you carefully read the previous paragraph, then perhaps you are interested in one phrase. "What is a trisection angle?" - for certain you will ask. The trisectrix is a bit like the bisectrix, but if you draw the latter, the angle will be divided into two equal parts, and in the construction of the trisection - by three. Naturally, the bisector of the angle is remembered more easily, because the trisection in the school is not taught. But for completeness, I'll tell you about it.

The trisectrix, as I said, can not be built only by a compass and a ruler, but it can be created with the help of the Fujita rules and some curves: Pascal's snail, quadratrix, conocoids of Nycomed, conic sections, Archimedes spiral.

Problems on the trisection of the angle are simply solved with the help of a non-pointer.

In geometry, there exists a theorem on trisectrixes of an angle. It is called the Morley theorem (Morley). She argues that the intersection points of the trisectrix of each corner in the middle will be the vertices of an equilateral triangle.

A small black triangle inside a large one will always be equilateral. This theorem was discovered by the British scientist Frank Morley in 1904.

That's how much you can learn about the separation of the angle: the trisectrix and the angle bisector always require detailed explanations. But there were a lot of definitions not yet revealed by me: the snail of Pascal, the conchoid Nycomed, etc. Do not doubt, you can write about them even more.