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Bisector of a triangle. Detailed theory with examples (2019)

Bisector of a triangle and its properties

Do you know what the midpoint of a segment is? Of course you do. What about the center of the circle? Same. What is the midpoint of an angle? You can say that this doesn't happen. But why can a segment be divided in half, but an angle cannot? It’s quite possible - just not a dot, but…. line.

Do you remember the joke: a bisector is a rat that runs around the corners and divides the corner in half. So, the real definition of a bisector is very similar to this joke:

Bisector of a triangle- this is the bisector segment of an angle of a triangle connecting the vertex of this angle with a point on the opposite side.

Once upon a time, ancient astronomers and mathematicians discovered many interesting properties of the bisector. This knowledge has greatly simplified people's lives. It has become easier to build, count distances, even adjust the firing of cannons... Knowledge of these properties will help us solve some GIA and Unified State Examination tasks!

The first knowledge that will help with this is bisector of an isosceles triangle.

By the way, do you remember all these terms? Do you remember how they differ from each other? No? Not scary. Let's figure it out now.

So, base of an isosceles triangle- this is the side that is not equal to any other. Look at the picture, which side do you think it is? That's right - this is the side.

The median is a line drawn from the vertex of a triangle and dividing the opposite side (that's it again) in half.

Notice we don't say, "Median of an isosceles triangle." Do you know why? Because a median drawn from a vertex of a triangle bisects the opposite side in ANY triangle.

Well, the height is a line drawn from the top and perpendicular to the base. You noticed? We are again talking about any triangle, not just an isosceles one. The height in ANY triangle is always perpendicular to the base.

So, have you figured it out? Almost. To understand even better and forever remember what a bisector, median and height are, you need to compare them with each other and understand how they are similar and how they differ from each other. At the same time, in order to remember better, it is better to describe everything in “human language”. Then you will easily operate in the language of mathematics, but at first you do not understand this language and you need to comprehend everything in your own language.

So, how are they similar? The bisector, the median and the altitude - they all “come out” from the vertex of the triangle and rest on the opposite side and “do something” either with the angle from which they come out, or with the opposite side. I think it's simple, no?

How are they different?

• The bisector divides the angle from which it emerges in half.
• The median divides the opposite side in half.
• The height is always perpendicular to the opposite side.

That's it. It's easy to understand. And once you understand, you can remember.

Now the next question. Why, in the case of an isosceles triangle, is the bisector both the median and the altitude?

You can simply look at the figure and make sure that the median divides into two absolutely equal triangles. That's all! But mathematicians do not like to believe their eyes. They need to prove everything. Scary word? Nothing like that - it's simple! Look: both have equal sides and, they generally have a common side and. (- bisector!) And so it turns out that two triangles have two equal sides and the angle between them. We recall the first sign of equality of triangles (if you don’t remember, look in the topic) and conclude that, and therefore = and.

This is already good - it means it turned out to be the median.

But what is it?

Let's look at the picture - . And we got it. So, too! Finally, hurray! And.

Did you find this proof a bit heavy? Look at the picture - two identical triangles speak for themselves.

In any case, remember firmly:

Now it’s more difficult: we’ll count angle between bisectors in any triangle! Don't be afraid, it's not that tricky. Look at the picture:

Let's count it. Do you remember that the sum of the angles of a triangle is?

Let's apply this amazing fact.

On the one hand, from:

That is.

Now let's look at:

But bisectors, bisectors!

Let's remember about:

Now through the letters

\angle AOC=90()^\circ +\frac(\angle B)(2)

Isn't it surprising? It turned out that the angle between the bisectors of two angles depends only on the third angle!

Well, we looked at two bisectors. What if there are three of them??!! Will they all intersect at one point?

Or will it be like this?

How do you think? So mathematicians thought and thought and proved:

Isn't that great?

Do you want to know why this happens?

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 all!

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 paragraph 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. 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, 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 sum

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

Theorem 1:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 6:

The bisector of a triangle is a common geometric concept that does not cause much difficulty in learning. Having knowledge about its properties, you can solve many problems 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 comes from the use of words in Latin, the meaning of which is “bi” - two, “sectio” - to cut. They specifically point to the geometric meaning of the concept - the division of space between the rays into two equal parts.

The bisector of a triangle is a segment that originates from the vertex 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 by students mathematical concepts use different terminology, which is reflected in poems or associations. Of course, using this definition is recommended for older children.

How is this line designated? Here we rely on the rules for designating segments or rays. If we're talking about about the designation of the angle bisector of a triangular figure, it is usually written as a segment whose ends are vertex and the point of intersection with the side opposite the vertex. Moreover, the beginning of the notation is written precisely from the vertex.

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, in school textbook you can find not so many properties of this geometric concept. The first property of the bisector of a triangle that 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:

1. 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.
2. In order to fit a circle into a triangular figure, it is necessary to determine the point at which these segments will intersect. This is the center point of the circle.
3. 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 remaining features into the system and present additional facts that will help to better understand the advantages of this geometric concept.

Length

One of the types of problems that cause difficulty for schoolchildren is finding the length of the bisector of an angle of a triangle. The first option, which contains its length, contains the following data:

• the amount of space between the rays from the vertex of which a given segment emerges;
• the lengths of the sides that form this angle.

To solve the problem formula used, the meaning of which is to find the ratio of the product of the values ​​of the sides that make up the angle, increased by 2 times, 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 a segment is drawn from angle A and intersects side BC at point K. We denote the value of A as Y. Based on this, AK = (2*AB*AC*cos(Y/2))/(AB+ AC).

The second version of the problem, in which the length of the bisector of a triangle is determined, contains the following data:

• the meanings of all sides of the figure are known.

When solving a problem of this type, initially determine the semi-perimeter. To do this, you need to add up the values ​​of all sides and divide in half: p=(AB+BC+AC)/2. Next, we apply the computational formula that 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 double root of the second power of the product of the lengths of the sides that are adjacent to the vertex by the semi-perimeter and the difference between the semi-perimeter and the length of the side opposite it to the sum of the sides that make up the angle. That is, AK = (2٦AB*AC*p*(p-BC))/(AB+AC).

Attention! To make it easier to master the material, you can turn to comic tales available on the Internet that tell about the “adventures” of this line.

One of the basics of geometry is finding the bisector, the ray that bisects an angle. The bisector of a triangle is the part of the bisector of any angle. This is a segment from the vertex of the angle to the intersection with the opposite side of the triangle.

If you draw bisectors from all angles, they will intersect at one point, which is called the center of the inscribed triangle.

You can calculate the bisector if you know the length of the side that it bisects, or the size of the angles of the triangle.

Bisector of an isosceles triangle

Since in an isosceles triangle two sides are equal to each other, then the bisectors of adjacent angles will be equal. Because The angles of the triangle are also equal.

When drawing a bisector from one of the corners, it will be considered the height of the given triangle and its median.

Problems of how to find the bisector of a triangle are solved using formulas.

To solve these formulas, the conditions must indicate the values ​​of the lengths of the sides, or the values ​​of the angles of the triangle. Knowing them, you can calculate the bisector using cosines or perimeter.

For example, take an isosceles triangle ABC and draw the bisector AE to the base BC. The resulting triangle AEB is right-angled. The bisector is its altitude, side AB is the hypotenuse of the right triangle, and BE and AE are the legs.

The Pythagorean theorem is applied - the square of the hypotenuse is equal to the sum of the squares of the legs. Based on it BE = v (AB - AE). Since AE is the median of triangle ABC, then side BE = BC/2. Thus BE = v(AB - (BC/4)).

If the base angle ABC is given, then the bisector of the triangle is AEB, AE = AB/sin(ABC). Base angle AEB, BAE = BAC/2. Therefore, the bisector AE = AB/cos (BAC/2).

How to find the bisector of a triangle inscribed in another triangle?

In an isosceles triangle ABC, draw side BC to side AC. This segment will be neither the bisector of the triangle nor its median. The Stewart formula applies here.

It is used to calculate the perimeter of a triangle - the sum of the lengths of all its sides. For ABC we calculate the semi-perimeter. This is the perimeter of the triangle divided in half.

P = (AB+ BC+ AC)/2. Using this formula, we calculate the bisector drawn to the side. VK = v(4*VS*AS*P (R-AV)/ (VS+AS).

By Stewart's theorem, you can also see that the bisector drawn to the other side of the triangle will be equal to VC, because these two sides of the triangle are equal to each other.

Bisector of a right triangle

In order to know how to find the bisector in a right triangle, you also need to use formulas. Do not forget that in a right triangle one angle is necessarily right, i.e. equal to 90 degrees. Thus, if the bisector starts from right angle, even if the condition does not indicate the sine or cosine of the angle, you can recognize them by the size of the angle.

• The bisector is found using Stewart's formula. If there is a triangle ABC, and its semi-perimeter is calculated as P = (AB+ BC+ AK)/2. Based on this, we calculate the bisector AE = v(4*VK*AK*P (P-AB)/ (VK+AK).
• The length of the bisector is determined in this way. AE = v (BK*AK) – (EB*EK), where EB and EK are the segments into which the bisector AE divides the side BK.
• Or you can use the cosines of the angles of a right triangle, if they are known. The bisector will be equal to (2*аb*(cos c/2))/(a+b).
• Or find the bisector like this. Using the formula (cos a) – (cos b)/2, find the divisor you need in the future. Next, the height drawn to side c is divided by the resulting value. To obtain cosines, you need to know the magnitude of the angles. Or calculate them based on the size of the only known angle - a right angle, 90 degrees.

Equilateral triangle

In such a triangle, all sides are equal to each other, and so are the angles. Therefore, all bisectors and medians will also be equal. If some of the side values ​​are unknown, then the value of one side will be needed. Because the sides are equal. And the sizes of the angles too. Therefore, to find the bisector using the cosine formula, you need to know or calculate the value of only one of the angles.

The length of the median and bisector of a triangle is equal to - L.

The sides of the triangle are equal - a.

In triangle ABC, bisector AE = (ABCv3)/2.

The same formula is used to calculate the height and median of an equilateral triangle.

Scalene triangle

In such a triangle, all sides have different meanings, therefore the bisectors are not equal to each other.

Take a triangle with arbitrary side values. If some values ​​of the sides are unknown, then they are calculated using the formula for the perimeter of a triangle.

After the angle bisectors have been drawn, it is worth adding a subscript1 to their designations. The segments into which the bisector divides the opposite side are also designated with the subscript 1.

The lengths of these segments are calculated using the sine theorem.

The length of the bisector is calculated as L = v ab – a1b1, where ab are the sides adjacent to the segments, and a1b1 is the product of the segments. The formula applies to all sides of a scalene triangle. The main thing is to know the lengths of the sides, or calculate them, knowing the values ​​of the adjacent angles.

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. External corner in a triangle the angle adjacent to internal corner 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 will find the length of the third side as the sum of the segments into which this side was divided by the bisector.

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

Then the entire segment AC = RC+ AR

AC = 3+2=5 cm.

Total ratings received: 108.

Mathematics, as we know, is the queen of sciences. It is no coincidence that teachers, especially those of the older generation, love this expression so much. Mathematics opens exclusively to those who know how, firstly, to think logically, and secondly, to those who always like to achieve an answer, operating with initial conditions, without cheating, but basing decisions on analysis, again building logical connections. These qualities, learned from school, can be modulated into adult serious life, both in work and in other difficult moments.

Today many people face problems in solving mathematical problems also in primary school.

However, even those schoolchildren who successfully master the primary mathematical program, moving to a new school and life stage, where algebra is separated from geometry, sometimes encounter serious difficulties. Meanwhile, having once learned and, most importantly, understood, how to find the bisector of a triangle, the student will forever remember this formula. Consider triangle ABC with three bisectors. As can be seen from the figure, they all converge at one point.

First, let us determine that the bisector of a triangle, and this is one of its most important properties, divides the angle from which such a segment originates in half. That is, in the example given, angle BAD equal to angle DAC.

Properties

1. The bisector of a triangle divides the side to which it is drawn into two segments that have the properties of proportionality to the sides that are adjacent to each segment, respectively. Thus, BD/CD = AB/AC.
2. Each triangle can have three given segments. Other significant properties concern both particular and general cases of the specific triangles under consideration.

Properties in isosceles triangles

Determination of the bisector of a triangle

Let us assume that in the triangle ABC under consideration, side AB = 5 cm, AC = 4 cm. Section CD = 3 cm.

Length Determination

The length can be determined by the following formula . AD= Square root from the difference between the product of sides and the product of proportional segments.

Find the length of side BC.

• From the properties it is known that BD/CD = AB/AC.
• So BD/CD = 5/4 = 1.25.
• BD/3 = 5/4.
• So BD = 3.75.
• ABxAC = 5×4=20.
• CDxBD = 3x3.75 = 11.25.

This example It is also intended to explicitly indicate the situation when the values ​​of the length of the bisector, like all other values ​​in mathematics, will not be expressed in natural numbers, but you should not be afraid of this.

Finding the Angle

To find the angles formed by a bisector, it is important, first of all, remember the sum of the angles, always 180 degrees. Let's assume that angle ABC is 70 degrees and angle BCA is 50 degrees. This means that through simple calculations we find that CAB = 180 – (70+50) = 60 degrees.

If we use the main property, according to which the corner from which she comes out, divides in half, we get equal values ​​of the angles BAD and CAD, each of which will be 60/2 = 30 degrees.

If additional clear example, consider a situation where only the angle BAD is known, equal to 28 degrees, and also the angle ABC, equal to 70 degrees. Using the property of the bisector, we immediately find the angle CAB by multiplying the value of the angle BAD by two. CAB = 28×2 =56. So, BAC = 180 – (70+56) or 180 – (70+28×2) = 180 – 126 = 54 degrees.

The situation when this segment acts as a median or height was not specifically considered, leaving other specialized articles for this.

Thus, we considered such a concept as triangle bisector, the formula for finding the length and angles of which is laid down and implemented in the given examples, which are intended to clearly show how it can be used to solve certain problems in geometry. Also related to this topic are concepts such as median and height. If this issue has become clear, one should turn to further study of various other properties of the triangle, without which further study of geometry is unthinkable.