The main elements of the triangle abc. What is the bisector of a triangle: properties related to the aspect ratio
Among the numerous subjects of the secondary school there is such as "geometry". It is traditionally believed that the founders of this systematic science are the Greeks. Today, Greek geometry is called elementary, since it was she who began the study of the simplest forms: planes, lines, and triangles. We will focus on the latter, or rather on the bisector of this figure. For those who have already forgotten, the bisector of a triangle is a segment of the bisector of one of the angles of the triangle, which divides it in half and connects the vertex to a point located on the opposite side.
The bisector of a triangle has a number of properties that you need to know when solving certain problems:
- The bisector of an angle is the locus of points that are equidistant from the sides adjacent to the angle.
- The bisector in a triangle divides the opposite side of the angle into segments that are proportional to the adjacent sides. For example, given triangle MKB, where a bisector emerges from angle K, connecting the vertex of this angle with point A on the opposite side of MB. Having analyzed this property and our triangle, we have MA/AB=MK/KB.
- The point at which the bisectors of all three angles of a triangle intersect is the center of a circle that is inscribed in the same triangle.
- The base of the bisectors of one external and two internal angles are on the same line, provided that the bisector of the external angle is not parallel to the opposite side of the triangle.
- If two bisectors of one then this
It should be noted that if three bisectors are given, then building a triangle using them, even with the help of a compass, is impossible.
Very often, when solving problems, the bisector of a triangle is unknown, but it is necessary to determine its length. To solve such a problem, it is necessary to know the angle that is divided by the bisector in half, and the sides adjacent to this angle. In this case, the desired length is defined as the ratio of the double product of the sides adjacent to the corner and the cosine of the angle divided in half to the sum of the sides adjacent to the corner. For example, given the same triangle MKB. The bisector leaves angle K and intersects the opposite side of MB at point A. The angle from which the bisector leaves is denoted by y. Now let's write down everything that is said in words in the form of a formula: KA = (2*MK*KB*cos y/2) / (MK+KB).
If the value of the angle from which the bisector of the triangle comes out is unknown, but all its sides are known, then to calculate the length of the bisector we will use an additional variable, which we will call the semi-perimeter and denote by the letter P: P=1/2*(MK+KB+MB). After that, we will make some changes to the previous formula, according to which the length of the bisector was determined, namely, in the numerator of the fraction we put twice the product of the lengths of the sides adjacent to the corner by the semiperimeter and the quotient, where the length of the third side is subtracted from the semiperimeter. We leave the denominator unchanged. In the form of a formula, it will look like this: KA=2*√(MK*KB*P*(P-MB)) / (MK+KB).
The bisector of an isosceles triangle, along with common properties, has several of its own. Let's remember what a triangle is. In such a triangle, two sides are equal, and the angles adjacent to the base are equal. It follows that the bisectors that descend to the sides of an isosceles triangle are equal to each other. In addition, the bisector lowered to the base is both the height and the median at the same time.
The interior angles of a triangle is called the bisector of the triangle.
The angle bisector of a triangle is also understood as the segment between its vertex and the point of intersection of the bisector with the opposite side of the triangle.
Theorem 8.
The three bisectors of a triangle intersect at one point.
Indeed, consider first the point Р of the intersection of two bisectors, for example, AK 1 and VC 2. This point is equally distant from the sides AB and AC, since it lies on the bisector of angle A, and is equally distant from the sides AB and BC, as belonging to the bisector of angle B. Therefore, it is equally distant from the sides AC and BC and thus belongs to the third bisector SK 3 , that is, at the point P all three bisectors intersect.
Properties of bisectors of internal and external angles of a triangle
Theorem 9.
The bisector of the interior angle of a triangle divides the opposite side into parts proportional to the adjacent sides. 
Proof. Consider the triangle ABC and the bisector of its angle B. Let us draw a straight line CM through the vertex C, parallel to the bisector BK, until it intersects at the point M as an extension of the side AB. Since VC is the bisector of the angle ABC, then ∠ ABK=∠ KBC. Further, ∠ ABK=∠ VMS, as the corresponding angles at parallel lines, and ∠ KBC=∠ VCM, as the cross-lying angles at parallel lines. Hence ∠ VCM=∠ VMS, and therefore the VMS triangle is isosceles, hence BC=VM. According to the theorem on parallel lines intersecting the sides of an angle, we have AK:K C=AB:VM=AB:BC, which was required to be proved.
Theorem 10
The bisector of the external angle B of the triangle ABC has a similar property: the segments AL and CL from the vertices A and C to the point L of the intersection of the bisector with the extension of the side AC are proportional to the sides of the triangle: AL: CL=AB :BC .
This property is proved in the same way as the previous one: an auxiliary straight line CM is drawn in the figure, parallel to the bisector BL . The angles BMC and BCM are equal, which means that the sides BM and BC of the triangle BMC are equal. From which we come to the conclusion AL:CL=AB:BC.
Theorem d4.
(the first formula for the bisector): If in triangle ABC the segment AL is the bisector of angle A, then AL? = AB AC - LB LC.
Proof: Let M be the point of intersection of the line AL with the circle circumscribed about the triangle ABC (Fig. 41). The BAM angle is equal to the MAC angle by convention. Angles BMA and BCA are equal as inscribed angles based on the same chord. Hence, triangles BAM and LAC are similar in two angles. Therefore, AL: AC = AB: AM. So AL AM = AB AC<=>AL (AL + LM) = AB AC<=>AL? = AB AC - AL LM = AB AC - BL LC. Which is what needed to be proven. Note: for the theorem on segments of intersecting chords in a circle and on inscribed angles, see the topic circle and circle.
Theorem d5.
(second formula for the bisector): In triangle ABC with sides AB=a, AC=b and angle A equal to 2? and the bisector l, the equality takes place:
l = (2ab / (a+b)) · cos?.
Proof: Let ABC be a given triangle, AL its bisector (Fig. 42), a=AB, b=AC, l=AL. Then S ABC = S ALB + S ALC . Hence absin2? = alsin? +blsin?<=>2absin? cos? = (a + b)lsin?<=>l = 2 (ab / (a+b)) cos?. The theorem has been proven.
What is the angle bisector of a triangle? To this question, some people have the notorious rat running around the corners and dividing the corner in half. "If the answer has to be "with humor", then perhaps it is correct. But from a scientific point of view, the answer to this question should have sounded something like this: 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 erroneous opinion. And what else is known about the angle bisector, besides its definition?
Like any locus of points, it has its own characteristics. The first of them is rather not even a sign, but a theorem that can be briefly expressed as follows: "If the opposite side is divided into two parts by a bisector, then their ratio will correspond to the ratio of the sides of a large triangle."
The second property that it has: the point of intersection of the bisectors of all angles is called the incenter.
The third sign: the bisectors of one internal and two external angles of a triangle intersect at the center of one of the three circles inscribed in it.
The fourth property of the angle bisector of a triangle is that if each of them is equal, then the last one is isosceles.
The fifth sign also concerns an isosceles triangle and is the main guideline for its recognition in the drawing by bisectors, namely: in an isosceles triangle, it simultaneously acts as a median and height.
An angle bisector can be constructed using a compass and straightedge:
The sixth rule says that it is impossible to construct a triangle using the latter only with the available bisectors, just as it is impossible to construct a doubling of a cube, a square of a circle and a trisection of an angle in this way. Strictly speaking, 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 the trisection of an angle?" - you will surely ask. The trisectrix is a bit similar to the bisector, but if you draw the latter, then the angle will be divided into two equal parts, and when constructing a trisection, into three. Naturally, the bisector of an angle is easier to remember, because the trisection is not taught at school. But for the sake of completeness, I will tell you about it.
The trisector, as I said, cannot be built only with a compass and a ruler, but it can be created using the Fujita rules and some curves: Pascal's snails, quadratics, Nicomedes conchoids, conic sections,
Problems on the trisection of an angle are quite simply solved with the help of nevsis.
In geometry, there is a theorem on the trisectors of an angle. It is called the Morley (Morley) theorem. She states that the points of intersection of the trisectors in the middle of each angle will be vertices
A small black triangle inside a large one will always be equilateral. This theorem was discovered by the British scientist Frank Morley in 1904.
Here's how much you can learn about the division of an angle: the trisector and bisector of an angle always require detailed explanations. But here many definitions have been given that have not yet been disclosed by me: Pascal's snail, Nicomedes' conchoid, etc. No doubt, more can be written about them.
PROPERTIES OF THE BISSECTOR
Bisector property: In a triangle, the bisector divides the opposite side into segments proportional to the adjacent sides.
Bisector of an external angle The bisector of an external angle of a triangle intersects the extension of its side at a point, the distances from which to the ends of this side are proportional, respectively, to the adjacent sides of the triangle. C B A D
Bisector length formulas:
The formula for finding the lengths of the segments into which the bisector divides the opposite side of the triangle
The formula for finding the ratio of the lengths of the segments into which the bisector is divided by the intersection point of the bisectors
Problem 1. One of the bisectors of a triangle is divided by the intersection point of the bisectors in a ratio of 3:2, counting from the vertex. Find the perimeter of a triangle if the length of the side of the triangle to which this bisector is drawn is 12 cm.
Solution We use the formula to find the ratio of the lengths of the segments into which the bisector is divided by the intersection point of the bisectors in the triangle: 30. Answer: P = 30cm.
Task 2 . Bisectors BD and CE ∆ ABC intersect at point O. AB=14, BC=6, AC=10. Find O D .
Solution. Let's use the formula for finding the length of the bisector: We have: BD = BD = = According to the formula for the ratio of the segments into which the bisector is divided by the intersection point of the bisectors: l = . 2 + 1 = 3 parts of everything.
this is part 1 OD = Answer: OD =
Problems In ∆ ABC, the bisectors AL and BK are drawn. Find the length of the segment KLif AB \u003d 15, AK \u003d 7.5, BL \u003d 5. In ∆ ABC, the bisector AD is drawn, and through point D is a straight line parallel to AC and intersecting AB at point E. Find the ratio of areas ∆ ABC and ∆ BDE , if AB = 5, AC = 7. Find the bisectors of acute angles of a right triangle with legs 24 cm and 18 cm. Bisector in a right triangle acute angle divides the opposite leg into segments 4 and 5 cm long. Determine the area of the triangle.
5. In an isosceles triangle, the base and side are 5 and 20 cm, respectively. Find the bisector of the angle at the base of the triangle. 6. Find the bisector of the right angle of a triangle whose legs are equal a and b. 7. Calculate the length of the bisector of angle A of triangle ABC with side lengths a = 18 cm, b = 15 cm, c = 12 cm. Find the ratio in which the bisectors of the interior angles divide at the point of their intersection.
Answers: Answer: Answer: Answer: Answer: Answer: Answer: Answer: Answer: AP = 6 AP = 10 see KL = CP =
The bisector of a triangle is a common geometric concept that does not cause much difficulty in learning. Knowing about its properties, many problems can be solved without much difficulty. What is a bisector? We will try to acquaint the reader with all the secrets of this mathematical line.
In contact with
The essence of the concept
The name of the concept came from the use of words in Latin, the meaning of which is "bi" - two, "sectio" - cut. They specifically point to the geometric meaning of the concept - breaking up the space between the rays into two equal parts.
The bisector of a triangle is a segment that originates from the top of the figure, and the other end is placed on the side that is located opposite it, while dividing the space into two identical parts.
Many teachers for quick associative memorization of mathematical concepts by students use different terminology, which is displayed in verses or associations. Of course, this definition is recommended for older children.
How is this line marked? Here we rely on the rules for designating segments or rays. If we are talking about the designation of the bisector of the angle of a triangular figure, then it is usually written as a segment, the ends of which are vertex and the point of intersection with the opposite side of the vertex. Moreover, the beginning of the designation is written exactly from the top.
Attention! How many bisectors does a triangle have? The answer is obvious: as many as there are vertices - three.
Properties
In addition to the definition, school textbook one can find not so many properties of this geometric concept. The first property of the bisector of a triangle, which schoolchildren are introduced to, is the inscribed center, and the second, directly related to it, is the proportionality of the segments. The bottom line is this:
- Whatever the dividing line, there are points on it that are at the same distance from the sides, which make up the space between the rays.
- In order to inscribe a circle in a triangular figure, it is necessary to determine the point at which these segments will intersect. This is the center point of the circle.
- Parts of a triangular side geometric figure, into which its dividing line divides, are in proportion to the sides forming the angle.
We will try to bring the rest of the features into a system and present additional facts that will help to better understand the merits of this geometric concept.
Length
One of the types of tasks that cause difficulty for schoolchildren is finding the length of the bisector of the angle of a triangle. The first option, in which its length is located, contains the following data:
- the size of the space between the rays, from the top of which the given segment emerges;
- the lengths of the sides that form this angle.
To solve the problem the formula is used, the meaning of which is to find the ratio of the doubled product of the values of the sides that make up the angle, by the cosine of its half, to the sum of the sides.
Let's look at a specific example. Suppose we are given a figure ABC, in which the segment is drawn from angle A and intersects side BC at point K. We denote the value of A by Y. Based on this, AK \u003d (2 * AB * AC * cos (Y / 2)) / (AB + AS).
The second version of the problem, in which the length of the bisector of a triangle is determined, contains the following data:
- the values of all sides of the figure are known.
When solving a problem of this type, initially determine the semiperimeter. To do this, add the values of all sides and divide in half: p \u003d (AB + BC + AC) / 2. Next, we apply the computational formula, which was used to determine the length of this segment in the previous problem. It is only necessary to make some changes to the essence of the formula in accordance with the new parameters. So, it is necessary to find the ratio of the twice root of the second degree from the product of the lengths of the sides that are adjacent to the top, to the semi-perimeter and the difference between the semi-perimeter and the length of the opposite side to the sum of the sides that make up the angle. That is, AK \u003d (2٦AB * AC * p * (r-BC)) / (AB + AC).
Attention! To make it easier to master the material, you can refer to the available on the Internet comic tales, telling about the "adventures" of this line.
