2 definition of angle bisector. What is the bisector of a triangle: properties related to the ratio of the sides

The word “bisector” is translated from French as “cutting in two.” The bisector of an angle is an “equidividing” angle, i.e. bisecting an angle.

Angle bisector - a ray drawn from the vertex of an angle between its sides and dividing the angle in half.

The bisector of an angle can be constructed using the degree measure of the angle using a protractor. To do this, the degree measure of a given angle is divided in half and the degree measure of a half angle is laid on one side of the vertex. The second side of such an angle will be the bisector of the given angle.

If a given angle has a degree measure of 60°, then the two angles constructed using the bisector are 30° each, since 60°:2 = 30°.

A straight angle is divided by a bisector into two right angles (180°:2=90°), any obtuse angle is divided by a bisector into two acute angles.

Constructing an angle bisector using a compass and ruler

To construct the bisector of an angle without a protractor, using only a compass and a ruler, you need to perform the following steps (see the figure above).

From the vertex of an angle, of any radius, it is necessary to draw an arc of a circle so that it intersects the sides of the angle
From each point (there are two) of the intersection of the arc and the side of the angle, again draw the soul of the circle (with a different radius)
Through any of the intersection points of the arcs of additionally constructed circles, draw a ray from the vertex of the angle, which will be the bisector of this angle

Angle bisector of a triangle

Angle bisector of a triangle is the segment of the angle bisector drawn from the vertex of the angle to its intersection with the opposite side.

A triangle has three bisectors, drawn from each of its vertices.

The bisector of the angle of a triangle has a lot of special properties, which are described in a separate article "

A bisector is a line that bisects an angle.

Did you encounter a bisector in the problem? Try to apply one (or sometimes several) of the following amazing properties.

1. Bisector in an isosceles triangle.

Aren't you afraid of the word "theorem"? If you are afraid, then it is in vain. Mathematicians are accustomed to calling a theorem any statement that can somehow be deduced from other, simpler statements.

So, attention, theorem!

Let's prove this theorem, that is, let us understand why this happens? Look at the isosceles.

Let's look at them carefully. And then we will see that

- general.

And this means (quickly remember the first sign of equality of triangles!) that.

So what? Do you want to say that? And the fact is that we have not yet looked at the third sides and the remaining angles of these triangles.

Now let's see. Once, then absolutely, and even in addition, .

So it turned out that

divided the side in half, that is, it turned out to be the median
, which means they are both like (look again at the picture).

So it turned out to be a bisector and a height too!

Hooray! We proved the theorem. But guess what, that’s not all. Also faithful converse theorem:

Proof? Are you really interested? Read the next level of theory!

And if you're not interested, then remember firmly:

Why remember this firmly? How can this help? But imagine that you have a task:

Given: .

Find: .

You immediately realize, bisector and, lo and behold, she divided the side in half! (according to condition...). If you firmly remember that this happens only in an isosceles triangle, then you draw a conclusion, which means, you write the answer: . Great, right? Of course, not all tasks will be so easy, but knowledge will definitely help!

And now the next property. Ready?

2. The bisector of an angle is the locus of points equidistant from the sides of the angle.

Scared? It's really no big deal. Lazy mathematicians hid four in two lines. So, what does it mean, “Bisector - locus of points"? This means that they are executed immediately twostatements:

If a point lies on a bisector, then the distances from it to the sides of the angle are equal.
If at some point the distances to the sides of the angle are equal, then this point Necessarily lies on the bisector.

Do you see the difference between statements 1 and 2? If not very much, then remember the Hatter from “Alice in Wonderland”: “So what else will you say, as if “I see what I eat” and “I eat what I see” are the same thing!”

So we need to prove statements 1 and 2, and then the statement: “a bisector is the locus of points equidistant from the sides of an angle” will be proven!

Why is 1 true?

Let's take any point on the bisector and call it .

Let us drop perpendiculars from this point to the sides of the angle.

And now...get ready to remember the signs of equality of right triangles! If you have forgotten them, then take a look at the section.

So...two right triangles: and. They have:

General hypotenuse.
(because it is a bisector!)

This means - by angle and hypotenuse. Therefore, the corresponding legs of these triangles are equal! That is.

We proved that the point is equally (or equally) distant from the sides of the angle. Point 1 is dealt with. Now let's move on to point 2.

Why is 2 true?

And let's connect the dots and.

This means that it lies on the bisector!

That's it!

How can all this be applied when solving problems? For example, in problems there is often the following phrase: “A circle touches the sides of an angle...”. Well, you need to find something.

Then you quickly realize that

And you can use equality.

3. Three bisectors in a triangle intersect at one point

From the property of a bisector to be the locus of points equidistant from the sides of an angle, the following statement follows:

How exactly does it come out? But look: two bisectors will definitely intersect, right?

And the third bisector could go like this:

But in reality, everything is much better!

Let's look at the intersection point of two bisectors. Let's call it .

What did we use here both times? Yes point 1, of course! If a point lies on a bisector, then it is equally distant from the sides of the angle.

And so it happened.

But look carefully at these two equalities! After all, it follows from them that and, therefore, .

And now it will come into play point 2: if the distances to the sides of an angle are equal, then the point lies on the bisector...what angle? Look at the picture again:

and are the distances to the sides of the angle, and they are equal, which means the point lies on the bisector of the angle. The third bisector passed through the same point! All three bisectors intersect at one point! And as an additional gift -

Radii inscribed circles.

(To be sure, look at another topic).

Well, now you'll never forget:

The point of intersection of the bisectors of a triangle is the center of the circle inscribed in it.

Let's move on to the next property... Wow, the bisector has many properties, right? And this is great, because the more properties, the more tools for solving bisector problems.

4. Bisector and parallelism, bisectors of adjacent angles

The fact that the bisector divides the angle in half in some cases leads to completely unexpected results. Here, for example,

Case 1

Great, right? Let's understand why this is so.

On the one hand, we draw a bisector!

But, on the other hand, there are angles that lie crosswise (remember the theme).

And now it turns out that; throw out the middle: ! - isosceles!

Case 2

Imagine a triangle (or look at the picture)

Let's continue the side beyond the point. Now we have two angles:

- internal corner
- the outer corner is outside, right?

So, and now someone wanted to draw not one, but two bisectors at once: both for and for. What will happen?

Will it work out? rectangular!

Surprisingly, this is exactly the case.

Let's figure it out.

What do you think the amount is?

Of course, - after all, they all together make such an angle that it turns out to be a straight line.

Now remember that and are bisectors and see that inside the angle there is exactly half from the sum of all four angles: and - - that is, exactly. You can also write it as an equation:

So, incredible but true:

The angle between the bisectors of the internal and external angles of a triangle is equal.

Case 3

Do you see that everything is the same here as for the internal and external corners?

Or let's think again why this happens?

Again, as for adjacent corners,

(as corresponding with parallel bases).

And again, they make up exactly half from the amount

Conclusion: If the problem contains bisectors adjacent angles or bisectors relevant angles of a parallelogram or trapezoid, then in this problem certainly a right triangle is involved, or maybe even a whole rectangle.

5. Bisector and opposite side

It turns out that the bisector of an angle of a triangle divides the opposite side not just in some way, but in a special and very interesting way:

That is:

An amazing fact, isn't it?

Now we will prove this fact, but get ready: it will be a little more difficult than before.

Again - exit to “space” - additional formation!

Let's go straight.

For what? We'll see now.

Let's continue the bisector until it intersects with the line.

Is this a familiar picture? Yes, yes, yes, exactly the same as in point 4, case 1 - it turns out that (- bisector)

Lying crosswise

So, that too.

Now let's look at the triangles and.

What can you say about them?

They are...similar. Well, yes, their angles are equal as vertical ones. So, in two corners.

Now we have the right to write the relations of the relevant parties.

And now in short notation:

Oh! Reminds me of something, right? Isn't this what we wanted to prove? Yes, yes, exactly that!

You see how great the “spacewalk” proved to be - the construction of an additional straight line - without it nothing would have happened! And so, we have proven that

Now you can safely use it! Let's look at one more property of the bisectors of the angles of a triangle - don't be alarmed, now the hardest part is over - it will be easier.

We get that

This knowledge can be applied in those problems where two bisectors are involved and only an angle is given, and the required quantities are maintained through or, conversely, given, but you need to find something involving the angle.

The basic knowledge about the bisector is over. By combining these facts, you will find the key to any bisector problem!

BISECTOR. SUMMARY AND BASIC FORMULAS

Theorem 1:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 6:

Inside an angle, equidistant from the sides of the angle.

Mnemonic rule

A bisector is a rat that runs around corners and bisects the corner.

Makes it easier to remember the wording. Most often used by children.

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Synonyms:

See what “Bisector” is in other dictionaries:

bisector- y, w. bissecrice f. math. A straight line passing through the vertex of an angle and dividing it in half. BAS 2. Draw a bisector. Vasyukova 1999. A bisector is a rat that runs around the corners and divides the corner in half. 1994. Belyanin. Lex. Brokg... ... Historical Dictionary of Gallicisms of the Russian Language

Mathematics, line, straight Dictionary of Russian synonyms. bisector noun, number of synonyms: 3rd line (182) ... Dictionary of synonyms

- (from Latin bis twice and seco I cut) an angle is a semi-straight line (ray), emanating from the vertex of the angle and dividing it in half... Big Encyclopedic Dictionary

- [ise], bisectors, female. (from lat. bissectrix secant across) (mat.). 1. In a corner there is a straight line dividing the angle in half. 2. In a triangle, a straight line drawn from some angle to the opposite side and dividing this side into parts is straight... ... Ushakov's Explanatory Dictionary

BISEXECTRISE, s, female. In mathematics: a ray (in 3 digits) emanating from the vertex of an angle and dividing it in half. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary

bisector- BISEXTER, s, f. Mathematics teacher at school. From school... Dictionary of Russian argot

bisector- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN mean line ... Technical Translator's Guide

BISECTOR- a ray emanating from the top of the angle and dividing it in half; any point B. is equally distant from the sides of the angle. The three B. angles of a triangle intersect at one point at the center of the circle inscribed in the triangle... Big Polytechnic Encyclopedia

- (French bissectrice lat. bis sectrix (bissectricis) cutting in two) geom. a ray passing through the vertex of an angle and dividing it in half. New dictionary of foreign words. by EdwART, 2009. bisector [ise], bisectors, w. [from Latin. bissectrix –… … Dictionary of foreign words of the Russian language

Y; and. [French bissecrice from lat. bis twice and secare dissect] Mat. A ray emerging from the top of a corner and dividing it in half. * * * bisector (from the Latin bis twice and seco I cut) of an angle, a half-line (ray) emanating from the vertex of the angle and dividing it... Encyclopedic Dictionary

Books

A bisector is such a rat..., Natalya Tsitronova. The author's first book is stories and essays about the dashing nineties... Written easily, with humor, without bloody or sex scenes...

The bisector of a triangle is a segment that divides an angle of a triangle into two equal angles. For example, if the angle of a triangle is 120 0, then by drawing a bisector, we will construct two angles of 60 0 each.

And since there are three angles in a triangle, three bisectors can be drawn. They all have one cut-off point. This point is the center of the circle inscribed in the triangle. In another way, this intersection point is called the incenter of the triangle.

When two bisectors of an internal and external angle intersect, an angle of 90 0 is obtained. An exterior angle in a triangle is the angle adjacent to the interior angle of a triangle.

Rice. 1. A triangle containing 3 bisectors

The bisector divides the opposite side into two segments that are connected to the sides:

$$(CL\over(LB)) = (AC\over(AB))$$

The bisector points are equidistant from the sides of the angle, which means that they are at the same distance from the sides of the angle. That is, if from any point of the bisector we drop perpendiculars to each of the sides of the angle of the triangle, then these perpendiculars will be equal..

If you draw a median, bisector and height from one vertex, then the median will be the longest segment, and the height will be the shortest.

Some properties of the bisector

In certain types of triangles, the bisector has special properties. This primarily applies to an isosceles triangle. This figure has two identical sides, and the third is called the base.

If you draw a bisector from the vertex of an angle of an isosceles triangle to the base, then it will have the properties of both height and median. Accordingly, the length of the bisector coincides with the length of the median and height.

Definitions:

Height- a perpendicular drawn from the vertex of a triangle to the opposite side.
Median– a segment that connects the vertex of a triangle and the middle of the opposite side.

Rice. 2. Bisector in an isosceles triangle

This also applies to an equilateral triangle, that is, a triangle in which all three sides are equal.

Example assignment

In triangle ABC: BR is the bisector, with AB = 6 cm, BC = 4 cm, and RC = 2 cm. Subtract the length of the third side.

Rice. 3. Bisector in a triangle

Solution:

The bisector divides the side of the triangle in a certain proportion. Let's use this proportion and express AR. Then we find the length of the third side as the sum of the segments into which the bisector divided this side.

$(AB\over(BC)) = (AR\over(RC))$
$RC=(6\over(4))*2=3 cm$

Then the entire segment AC = RC+ AR

AC = 3+2=5 cm.

In an isosceles triangle, the bisector drawn to the base divides the triangle into two equal right triangles.

What have we learned?

After studying the topic of bisector, we learned that it divides an angle into two equal angles. And if you draw it in an isosceles or equilateral triangle to the base, then it will have the properties of both medians and heights at the same time.

Test on the topic

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What is the bisector of an angle?

A besector is a rat that walks around corners and divides the corner in half
Properties of bisectors

a2a1=cb
la=c+bcb(b+c+a)(b+ca)
la=c+b2bc cos2
la=hacos2
la=bca1a2
Where:
like this somehow))
The besector of a straight angle divides it into 2 right angles
it's a rat that divides into pieces
Bisector (from Latin bi-double, and sectio cutting) of an angle is a ray with a beginning at the vertex of the angle, dividing the angle into two equal parts.
Bisector (from Latin bi-double, and sectio cutting) of an angle is a ray with a beginning at the vertex of the angle, dividing the angle into two equal parts.
A bisector is a rat that runs around the corners and divides the corner into genders
ray dividing an angle into 2 equal angles
A bisector is a rat that runs around the corners and divides the corner in half!
😉
Bisector (from Latin bi-double, and sectio cutting) of an angle is a ray with a beginning at the vertex of the angle, dividing the angle into two equal parts.
The bisector of an angle (together with its extension) is the locus of points equidistant from the sides of the angle (or their extensions).
Definition. The bisector of an angle of a triangle is the bisector segment of that angle connecting that vertex to a point on the opposite side.
Any of the three bisectors of the interior angles of a triangle is called a triangle bisector.
The bisector of an angle of a triangle can mean one of two things: the ray bisector of this angle or the segment of the bisector of this angle before its intersection with the side of the triangle.
Properties of bisectors
The bisector of an angle of a triangle divides the opposite side in a ratio equal to the ratio of the two adjacent sides.
The bisectors of the interior angles of a triangle intersect at one point. This point is called the center of the inscribed circle.
The bisectors of the internal and external angles are perpendicular.
If the bisector of an exterior angle of a triangle intersects the extension of the opposite side, then ADBD=ACBC.
The bisectors of one internal and two external angles of a triangle intersect at one point. This point is the center of one of the three excircles of this triangle.
The bases of the bisectors of two internal and one external angles of a triangle lie on the same straight line if the bisector of the external angle is not parallel to the opposite side of the triangle.
If the bisectors of the external angles of a triangle are not parallel to opposite sides, then their bases lie on the same straight line.
a2a1=cb
la=c+bcb(b+c+a)(b+c#8722;a)
la=c+b2bc cos2
la=hacos2#8722;
la=bc#8722;a1a2
Where:
la bisector drawn to side a,
a, b, c sides of the triangle against vertices A, B, C respectively,
al,a 2 segments into which the bisector lc divides side c,
interior angles of a triangle at vertices a, b, c, respectively,
ha is the height of the triangle dropped to side a.
bisector is a line that divides an angle into divisions
Bisector (from Latin bi-double, and sectio cutting) of an angle is a ray with a beginning at the vertex of the angle, dividing the angle into two equal parts.
The bisector of an angle (together with its extension) is the locus of points equidistant from the sides of the angle (or their extensions).
A bisector is a rat that walks around the corners, dividing the corner in half
bisector, such a rat, runs around the corners and divides the corner with hits)
Bisects an angle
the line that divides it (the angle) in half.
A bisector is a rat that runs around the corners and divides them in half