Triangle . Acute, obtuse and right triangle.

Legs and hypotenuse. Isosceles and equilateral triangle.

Sum of angles of a triangle.

External angle of a triangle. Signs of equality of triangles.

Remarkable lines and points in a triangle: heights, medians,

bisectors, median e perpendiculars, orthocenter,

center of gravity, center of a circumscribed circle, center of an inscribed circle.

Pythagorean theorem. Aspect ratio in an arbitrary triangle.

Triangle is a polygon with three sides (or three angles). The sides of a triangle are often indicated by small letters that correspond to the capital letters representing the opposite vertices.

If all three angles are acute (Fig. 20), then this acute triangle . If one of the angles is right(C, Fig.21), that is right triangle; sidesa, bforming a right angle are called legs; sidecopposite the right angle is called hypotenuse. If one of obtuse angles (B, Fig. 22), that is obtuse triangle.


Triangle ABC (Fig. 23) - isosceles, If two its sides are equal (a= c); these equal sides are called lateral, the third party is called basis triangle. Triangle ABC (Fig. 24) – equilateral, If All its sides are equal (a = b = c). In general ( a ≠ b ≠ c) we have scalene triangle .

Basic properties of triangles. In any triangle:

1. Opposite the larger side lies the larger angle, and vice versa.

2. Equal angles lie opposite equal sides, and vice versa.

In particular, all angles in equilateral triangle are equal.

3. The sum of the angles of a triangle is 180 º .

From the last two properties it follows that every angle in an equilateral

triangle is 60 º.

4. Continuing one of the sides of the triangle (AC, Fig. 25), we get external

angle BCD . The external angle of a triangle is equal to the sum of the internal angles,

not adjacent to it : BCD = A + B.

5. Any side of a triangle is less than the sum of the other two sides and greater

their differences (a < b + c, a > b – c;b < a + c, b > a – c;c < a + b,c > a – b).

Signs of equality of triangles.

Triangles are congruent if they are respectively equal:

a ) two sides and the angle between them;

b ) two corners and the side adjacent to them;

c) three sides.

Signs of equality of right triangles.

Two rectangular triangles are equal if one of the following conditions is true:

1) their legs are equal;

2) the leg and hypotenuse of one triangle are equal to the leg and hypotenuse of the other;

3) the hypotenuse and acute angle of one triangle are equal to the hypotenuse and acute angle of the other;

4) the leg and the adjacent acute angle of one triangle are equal to the leg and the adjacent acute angle of the other;

5) the leg and the opposite acute angle of one triangle are equal to the leg and the opposite acute angle of the other.

Wonderful lines and points in the triangle.

Height triangle isperpendicular,lowered from any vertex to the opposite side ( or its continuation). This side is calledbase of the triangle . The three altitudes of a triangle always intersectat one point, called orthocenter triangle. Orthocenter of an acute triangle (point O , Fig. 26) is located inside the triangle, andorthocenter of an obtuse triangle (point O , fig.27) – outside; The orthocenter of a right triangle coincides with the vertex of the right angle.

Median - This line segment , connecting any vertex of a triangle to the middle of the opposite side. Three medians of a triangle (AD, BE, CF, fig.28) intersect at one point O , always lying inside the triangle and being his center of gravity. This point divides each median in a ratio of 2:1, counting from the vertex.

Bisector - This bisector segment angle from vertex to point intersections with the opposite side. Three bisectors of a triangle (AD, BE, CF, fig.29) intersect at one point Oh, always lying inside the triangle And being center of the inscribed circle(see section “Inscribedand circumscribed polygons").

The bisector divides the opposite side into parts proportional to the adjacent sides ; for example, in Fig. 29 AE: CE = AB: BC.

Median perpendicular is a perpendicular drawn from the middle segment points (sides). Three perpendicular bisectors of triangle ABC(KO, MO, NO, Fig. 30 ) intersect at one point O, which is center circumscribed circle (points K, M, N – the midpoints of the sides of the triangle ABC).

In an acute triangle, this point lies inside the triangle; in obtuse - outside; in a rectangular - in the middle of the hypotenuse. Orthocenter, center of gravity, circumcenter and inscribed circle coincide only in an equilateral triangle.

Pythagorean theorem. In a right triangle, the square of lengthThe hypotenuse is equal to the sum of the squares of the lengths of the legs.

The proof of the Pythagorean theorem follows clearly from Fig. 31. Consider a right triangle ABC with legs a, b and hypotenuse c.

Let's build a square AKMB using the hypotenuse AB as a side. Thencontinue the sides of the right triangle ABC so as to get a square CDEF , whose side is equala + b .Now it is clear that the area of ​​the square CDEF is equal to ( a+b) 2 . On the other hand, this area equals the sum areas four right triangles and the square AKMB, that is

c 2 + 4 (ab / 2) = c 2 + 2 ab,

from here,

c 2 + 2 ab= (a+b) 2 ,

and finally we have:

c 2 =a 2 +b 2 .

Aspect ratio in an arbitrary triangle.

In the general case (for an arbitrary triangle) we have:

c 2 =a 2 +b 2 – 2ab· cos C,

where C – angle between sidesa And b .

Standard designations

Triangle with vertices A, B And C is designated as (see figure). A triangle has three sides:

The lengths of the sides of a triangle are indicated by lowercase Latin letters (a, b, c):

A triangle has the following angles:

The angle values ​​at the corresponding vertices are traditionally denoted by Greek letters (α, β, γ).

Signs of equality of triangles

A triangle on the Euclidean plane can be uniquely determined (up to congruence) by the following triplets of basic elements:

1. a, b, γ (equality on two sides and the angle lying between them);
2. a, β, γ (equality on the side and two adjacent angles);
3. a, b, c (equality on three sides).

Signs of equality of right triangles:

1. along the leg and hypotenuse;
2. on two legs;
3. along the leg and acute angle;
4. along the hypotenuse and acute angle.

Some points in the triangle are “paired”. For example, there are two points from which all sides are visible either at an angle of 60° or at an angle of 120°. They're called Torricelli dots. There are also two points whose projections onto the sides lie at the vertices of a regular triangle. This - Apollonius points. Points and such are called Brocard points.

Direct

In any triangle, the center of gravity, the orthocenter and the center of the circumcircle lie on the same straight line, called Euler's line.

The straight line passing through the center of the circumcircle and the Lemoine point is called Brocard axis. The Apollonius points lie on it. The Torricelli point and the Lemoine point also lie on the same line. The bases of the external bisectors of the angles of a triangle lie on one straight line called axis of external bisectors. The intersection points of the lines containing the sides of an orthotriangle with the lines containing the sides of the triangle also lie on the same line. This line is called orthocentric axis, it is perpendicular to the Euler straight line.

If we take a point on the circumcircle of a triangle, then its projections onto the sides of the triangle will lie on the same straight line, called Simson's straight this point. Simson's lines of diametrically opposite points are perpendicular.

Triangles

• A triangle with vertices at the bases drawn through a given point is called cevian triangle this point.
• A triangle with vertices in the projections of a given point onto the sides is called sod or pedal triangle this point.
• A triangle with vertices at the second points of intersection of lines drawn through the vertices and a given point with the circumscribed circle is called circumferential triangle. The circumferential triangle is similar to the sod triangle.

Circles

• Inscribed circle- a circle touching all three sides of the triangle. She's the only one. The center of the inscribed circle is called incenter.
• Circumcircle- a circle passing through all three vertices of a triangle. The circumscribed circle is also unique.
• Excircle- a circle touching one side of the triangle and the continuation of the other two sides. There are three such circles in a triangle. Their radical center is the center of the inscribed circle of the medial triangle, called Spiker's point.

The midpoints of the three sides of a triangle, the bases of its three altitudes and the midpoints of the three segments connecting its vertices with the orthocenter lie on one circle called circle of nine points or Euler circle. The center of the nine-point circle lies on the Euler line. A circle of nine points touches an inscribed circle and three excircles. The point of tangency between the inscribed circle and the circle of nine points is called Feuerbach point. If from each vertex of the triangle we lay outwards on straight lines containing the sides, orthoses equal in length to the opposite sides, then the resulting six points lie on the same circle - Conway circle. Three circles can be inscribed in any triangle in such a way that each of them touches two sides of the triangle and two other circles. Such circles are called Malfatti circles. The centers of the circumscribed circles of the six triangles into which the triangle is divided by medians lie on one circle, which is called circumference of Lamun.

A triangle has three circles that touch two sides of the triangle and the circumcircle. Such circles are called semi-inscribed or Verrier circles. The segments connecting the points of tangency of the Verrier circles with the circumcircle intersect at one point called Verrier's point. It serves as the center of a homothety, which transforms a circumcircle into an inscribed circle. The points of contact of the Verrier circles with the sides lie on a straight line that passes through the center of the inscribed circle.

The segments connecting the points of tangency of the inscribed circle with the vertices intersect at one point called Gergonne point, and the segments connecting the vertices with the points of tangency of the excircles are in Nagel point.

Ellipses, parabolas and hyperbolas

Inscribed conic (ellipse) and its perspector

An infinite number of conics (ellipses, parabolas or hyperbolas) can be inscribed into a triangle. If we inscribe an arbitrary conic into a triangle and connect the tangent points with opposite vertices, then the resulting straight lines will intersect at one point called prospector bunks. For any point of the plane that does not lie on a side or on its extension, there is an inscribed conic with a perspector at this point.

The described Steiner ellipse and the cevians passing through its foci

An ellipse can be inscribed in a triangle, touching the sides in the middle. Such an ellipse is called inscribed Steiner ellipse(its perspective will be the centroid of the triangle). The described ellipse, which touches the lines passing through the vertices parallel to the sides, is called described by the Steiner ellipse. If we transform a triangle into a regular triangle using an affine transformation (“skew”), then its inscribed and circumscribed Steiner ellipse will transform into an inscribed and circumscribed circle. The Chevian lines drawn through the foci of the described Steiner ellipse (Scutin points) are equal (Scutin's theorem). Of all the described ellipses, the described Steiner ellipse has the smallest area, and of all the inscribed ellipses, the inscribed Steiner ellipse has the largest area.

Brocard ellipse and its perspector - Lemoine point

An ellipse with foci at Brocard points is called Brocard ellipse. Its perspective is the Lemoine point.

Properties of an inscribed parabola

Kiepert parabola

The prospects of the inscribed parabolas lie on the described Steiner ellipse. The focus of an inscribed parabola lies on the circumcircle, and the directrix passes through the orthocenter. A parabola inscribed in a triangle and having Euler's directrix as its directrix is ​​called Kiepert parabola. Its perspector is the fourth point of intersection of the circumscribed circle and the circumscribed Steiner ellipse, called Steiner point.

Kiepert's hyperbole

If the described hyperbola passes through the point of intersection of the heights, then it is equilateral (that is, its asymptotes are perpendicular). The intersection point of the asymptotes of an equilateral hyperbola lies on the circle of nine points.

Transformations

If the lines passing through the vertices and some point not lying on the sides and their extensions are reflected relative to the corresponding bisectors, then their images will also intersect at one point, which is called isogonally conjugate the original one (if the point lay on the circumscribed circle, then the resulting straight lines will be parallel). Many pairs of remarkable points are isogonally conjugate: the circumcenter and the orthocenter, the centroid and the Lemoine point, the Brocard points. The Apollonius points are isogonally conjugate to the Torricelli points, and the center of the inscribed circle is isogonally conjugate to itself. Under the action of isogonal conjugation, straight lines transform into circumscribed conics, and circumscribed conics into straight lines. Thus, the Kiepert hyperbola and the Brocard axis, the Jenzabek hyperbola and the Euler straight line, the Feuerbach hyperbola and the line of centers of the inscribed and circumscribed circles are isogonally conjugate. The circumcircles of the triangles of isogonally conjugate points coincide. The foci of inscribed ellipses are isogonally conjugate.

If, instead of a symmetrical cevian, we take a cevian whose base is as distant from the middle of the side as the base of the original one, then such cevians will also intersect at one point. The resulting transformation is called isotomic conjugation. It also converts straight lines into described conics. The Gergonne and Nagel points are isotomically conjugate. Under affine transformations, isotomically conjugate points are transformed into isotomically conjugate points. With isotomic conjugation, the described Steiner ellipse will go into the infinitely distant straight line.

If in the segments cut off by the sides of the triangle from the circumcircle, we inscribe circles touching the sides at the bases of the cevians drawn through a certain point, and then connect the tangent points of these circles with the circumcircle with opposite vertices, then such straight lines will intersect at one point. A plane transformation that matches the original point to the resulting one is called isocircular transformation. The composition of isogonal and isotomic conjugates is the composition of an isocircular transformation with itself. This composition is a projective transformation, which leaves the sides of the triangle in place, and transforms the axis of the external bisectors into a straight line at infinity.

If we extend the sides of a Chevian triangle of a certain point and take their points of intersection with the corresponding sides, then the resulting points of intersection will lie on one straight line, called trilinear polar starting point. The orthocentric axis is the trilinear polar of the orthocenter; the trilinear polar of the center of the inscribed circle is the axis of the external bisectors. Trilinear polars of points lying on a circumscribed conic intersect at one point (for a circumscribed circle this is the Lemoine point, for a circumscribed Steiner ellipse it is the centroid). The composition of an isogonal (or isotomic) conjugate and a trilinear polar is a duality transformation (if a point isogonally (isotomically) conjugate to a point lies on the trilinear polar of a point, then the trilinear polar of a point isogonally (isotomically) conjugate to a point lies on the trilinear polar of a point).

Cubes

Ratios in a triangle

Note: in this section, , , are the lengths of the three sides of the triangle, and , , are the angles lying respectively opposite these three sides (opposite angles).

Triangle inequality

In a non-degenerate triangle, the sum of the lengths of its two sides is greater than the length of the third side, in a degenerate triangle it is equal. In other words, the lengths of the sides of a triangle are related by the following inequalities:

The triangle inequality is one of the axioms of metrics.

Triangle Angle Sum Theorem

Theorem of sines

,

where R is the radius of the circle circumscribed around the triangle. It follows from the theorem that if a< b < c, то α < β < γ.

Cosine theorem

Tangent theorem

Other ratios

Metric ratios in a triangle are given for:

Solving triangles

Calculating the unknown sides and angles of a triangle based on the known ones has historically been called “solving triangles.” The above general trigonometric theorems are used.

Area of ​​a triangle

Special cases Notation

For the area the following inequalities are valid:

Calculating the area of ​​a triangle in space using vectors

Let the vertices of the triangle be at points , , .

Let's introduce the area vector . The length of this vector is equal to the area of ​​the triangle, and it is directed normal to the plane of the triangle:

Let us set , where , , are the projections of the triangle onto the coordinate planes. Wherein

and similarly

The area of ​​the triangle is .

An alternative is to calculate the lengths of the sides (using the Pythagorean theorem) and then using Heron's formula.

Triangle theorems

This section is not completed.

Triangle- this is a figure consisting of three points and three segments, while the three points do not lie on the same line, but three segments connect these points in pairs. To be more precise, the points of a triangle are called its vertices, and the segments are called sides. A triangle is designated by its vertices, and instead of the long word triangle, the symbol Δ is drawn.

Let's now take a closer look at the types of triangles.

1. An isosceles triangle is a triangle that has two identical sides, which are also called lateral, the third side, different from those two, is called the base.
2. An equilateral triangle is a triangle with equal sides, also sometimes called a regular triangle.
3. A right triangle is a triangle that has a right angle (90 degrees).
4. An acute triangle is a triangle in which all angles are acute (that is, less than 90 degrees).
5. An obtuse triangle is a triangle in which one of the angles is obtuse (that is, more than 90 degrees).

In principle, it is easy to remember the features of each type of triangle, so which names speak for themselves.

Take, for example, triangle ABC. A, B, C are its vertices, and AB, BC and AC are its sides, respectively.

Now let's look at the structure of this triangle in more detail. The angle of triangle ABC at vertex A is the angle formed by half lines AB and AC. Similarly, we can determine the angles that lie at vertex B and at vertex C.

The altitude of a triangle is the perpendicular that descends from a given vertex to the line that is opposite the vertex.

The bisector of a triangle is the bisector segment of an angle of a given triangle that connects a vertex to a point on the opposite side.

The median of a triangle, which is drawn from a given vertex, is a segment connecting this vertex with the midpoint of the opposite side of the triangle.

The midline of a triangle is a segment that connects the midpoints of two sides of a given triangle. This designation also has a certain theorem, which says that the middle line of a triangle is always parallel to the third side, and is also equal to half of it.

All these notations (median, bisector, height, midline of a triangle) will definitely be needed in solving practical problems. Moreover, without knowing the properties of these vertices, you are unlikely to be able to solve any problem related to triangles.

on sides a, b, c respectively;
Cosine theorem

a

2

= b

2

+c

2

- 2bccosα

a+c a-c	=	tg	α + γ; 2	=	ctg	β 2
		tg	α - γ 2		tg	α - γ 2

R=

c 2

= m

c


Equilateral triangle

S=

If three points that do not lie on the same line are connected by segments, we get a triangle. One of the sides of a triangle is often called its base.

Theorem. The sum of the angles of a triangle is 180 0

If all three angles of a triangle are acute, then the triangle is called acute-angled.

If one of the angles of a triangle is obtuse, then the triangle is called obtuse-angled.

If one of the angles of a triangle is right, then the triangle is called rectangular. The side of a right triangle opposite the right angle is called hypotenuse, and the other two sides are legs.

In any triangle, the larger angle lies opposite the larger side; opposite equal sides - equal angles, and vice versa. Any side of a triangle is less than the sum of the other two sides, and also greater than the difference of the other two sides.

Continuing one of the sides of the triangle, we get an external angle. Angle ABD - external.

Signs of equality of triangles

If two triangles are congruent, then the elements (sides and angles) of one triangle are respectively equal to the elements of the other triangle.

Theorem. Two triangles are equal if two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of the other.

Theorem. Two triangles are equal if a side and two adjacent angles of one triangle are equal to the side and two adjacent angles of the other, respectively.

Theorem. Two triangles are congruent if three sides of one triangle are respectively equal to three sides of the other.

Median, bisector and height of a triangle

The segment connecting the vertex of a triangle with the midpoint of the opposite side is called median triangle.

A ray emanating from the vertex of an angle and dividing it into two equal angles is called bisector. The bisector divides the opposite side into parts proportional to the sides adjacent to it.

The perpendicular drawn from the vertex of a triangle to the line containing the opposite side is called height triangle.

Remarkable points of the triangle. 1) The bisectors of a triangle intersect at one point.

2) The perpendicular bisectors to the sides of the triangle intersect at one point.

3) The altitudes of the triangle (or their extensions) intersect at one point.

4) The medians of a triangle intersect at one point.

Isosceles triangle

A triangle is called isosceles if its two sides are equal. Equal sides are called sides, and the third party - basis isosceles triangle.

A triangle in which all sides are equal is called equilateral.

Theorem. In an isosceles triangle, the base angles are equal.

Theorem. In an isosceles triangle, the bisector drawn to the base is the median and altitude.

The science of geometry tells us what a triangle, square, and cube are. In the modern world, everyone without exception studies it in schools. Also, the science that studies directly what a triangle is and what properties it has is trigonometry. She explores in detail all phenomena related to data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems associated with them.

What is a triangle? Definition

This is a flat polygon. It has three corners, as is clear from its name. It also has three sides and three vertices, the first of them are segments, the second are points. Knowing what two angles are equal to, you can find the third by subtracting the sum of the first two from the number 180.

What types of triangles are there?

They can be classified according to various criteria.

First of all, they are divided into acute-angled, obtuse-angled and rectangular. The former have acute angles, that is, those that are equal to less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. Acute triangles also include equilateral triangles. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.

Right triangle

It is impossible not to talk about what a right triangle is.

Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The remaining two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. Using it, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can also recall an isosceles triangle. This is one in which two of the sides are equal, and two angles are also equal.

What are leg and hypotenuse?

A leg is one of the sides of a triangle that forms an angle of 90 degrees. The hypotenuse is the remaining side that is opposite the right angle. You can lower a perpendicular from it onto the leg. The ratio of the adjacent side to the hypotenuse is called cosine, and the opposite side is called sine.

- what are its features?

It's rectangular. Its legs are three and four, and its hypotenuse is five. If you see that the legs of a given triangle are equal to three and four, you can rest assured that the hypotenuse will be equal to five. Also, using this principle, you can easily determine that the leg will be equal to three if the second is equal to four, and the hypotenuse is equal to five. To prove this statement, you can apply the Pythagorean theorem. If two legs are equal to 3 and 4, then 9 + 16 = 25, the root of 25 is 5, that is, the hypotenuse is equal to 5. An Egyptian triangle is also a right triangle whose sides are 6, 8 and 10; 9, 12 and 15 and other numbers with the ratio 3:4:5.

What else could a triangle be?

Triangles can also be inscribed or circumscribed. The figure around which the circle is described is called inscribed; all its vertices are points lying on the circle. A circumscribed triangle is one into which a circle is inscribed. All its sides come into contact with it at certain points.

How is it located?

The area of ​​any figure is measured in square units (sq. meters, sq. millimeters, sq. centimeters, sq. decimeters, etc.) This value can be calculated in a variety of ways, depending on the type of triangle. The area of ​​any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from the opposite corner, and dividing this figure by two. You can also find this value by multiplying two sides. Then multiply this number by the sine of the angle located between these sides, and divide this result by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this you need to find half the perimeter. Then alternately subtract different sides from this number and multiply the resulting four values. Next, find from the number that came out. The area of ​​an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by that circumscribed around it, multiplied by four.

The area of ​​a circumscribed triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: square the side, multiply the resulting figure by the root of three, then divide this number by four. In a similar way, you can calculate the height of a triangle in which all sides are equal; to do this, you need to multiply one of them by the root of three, and then divide this number by two.

Theorems related to triangle

The main theorems that are associated with this figure are the Pythagorean theorem described above and cosines. The second (of sines) is that if you divide any side by the sine of the angle opposite it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosines) is that if from the sum of the squares of the two sides we subtract their product multiplied by two and the cosine of the angle located between them, then we get the square of the third side.

Dali Triangle - what is it?

Many, when faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. The Dali Triangle is the common name for three places that are closely connected with the life of the famous artist. Its “peaks” are the house in which Salvador Dali lived, the castle that he gave to his wife, as well as the museum of surrealist paintings. During a tour of these places you can learn many interesting facts about this unique creative artist, known all over the world.