What is a triangle angle bisector? To this question, for some people, a notorious saying breaks their tongue : "This is a rat running in corners and dividing the angle in half." If the answer should be "with humor", then perhaps it is correct. But from a scientific point of view, the answer to this question would sound 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 bisector until it intersects with the opposite side of the triangle. This is not an erroneous opinion. And what else is known about the bisector of the angle, except for its definition?
Like any geometric place of points, it has its own characteristics. The first of them is rather, not even a sign, but a theorem that can be summarized as follows: "If the bisector is divided into two sides, then their ratio will correspond to the ratio of the sides of a large triangle."
The second property that it has: the intersection point of the bisectors of all angles is called an incenter.
The third sign: the bisectors of one inner and two outer corners of the triangle intersect in the center of one of the three inscribed circles in it.
The fourth property of the bisector of the angle of a triangle is that if each of them is equal, then the latter is isosceles.
The fifth feature also relates to 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 built using a compass and a ruler:
The sixth rule states that it is impossible to construct a triangle with the help of the latter only with the existing bisectors, as it is impossible to construct a doubling of the cube, the square of the circle and the trisection of the angle in this way. Actually, this is all the properties of the bisector of the angle of a triangle.
If you carefully read the previous paragraph, then perhaps you were interested in one phrase. "What is a trisection of an angle?" - surely you ask. The trisector is a bit similar to the bisector, but if you draw the latter, the angle will be divided into two equal parts, and when building a trisection, it will be divided into three. Naturally, the angle bisector is easier to remember, because trisection is not taught at school. But for completeness, I’ll tell you about her.
The trisectrix, as I already said, cannot be built only with a compass and a ruler, but it can be created using the rules of Fujita and some curves: snails of Pascal, quadrice, conchoid Nycomed, conical sections, spiral of Archimedes.
The tasks of trisection of an angle are quite easily solved with the help of nevisis.
In geometry, there is a theorem on the trisector of an angle. It is called the Morley (Morley) theorem. She claims that the intersection points in the middle of the trisectors of each angle will be the vertices of an equilateral triangle.
The small black triangle inside the large will always be equilateral. This theorem was discovered by British scientist Frank Morley in 1904.
Here's how much you can learn about angle separation: the trisector and bisector of an angle always require detailed explanations. But here a lot of definitions have not yet been revealed by me: snail Pascal, conchoid Nycomed, etc. Do not hesitate to write more about them.