The solution of the right triangle. Equilateral triangle

Proper triangle, R. - radius of the described circle, r. - radius inscribed circle.

The radius of the inscribed circle of the correct triangle, expressed through his side:

r \u003d \\ FRAC (\\ SQRT 3) (6) A

The radius of the described circumference of the correct triangle, expressed through his side:

R \u003d \\ FRAC (\\ SQRT 3) (3) a

Perimeter of the right triangle:

P \u003d 3A \u003d 3 \\ SQRT 3 R \u003d 6 \\ SQRT 3 R

Heights, medians and bisector of the right triangle:

h \u003d M \u003d L \u003d \\ FRAC (\\ SQRT 3) (2) A

The area of \u200b\u200bthe right triangle is calculated by the formulas:

S \u003d \\ FRAC (\\ SQRT 3) (4) a ^ 2 \u003d \\ FRAC (3 \\ SQRT 3) (4) R ^ 2 \u003d 3 \\ SQRT 3 R ^ 2 \u003d \\ FRAC (\\ SQRT 3) (36) P ^ 2.

The radius of the circumference described is equal to the double radius of the inscribed circle:

R \u003d 2r.

Proper triangles can be coated plane.

In the right triangle, the circumference of nine points coincides with the inscribed circle.

For an equilateral triangle T, a group of movements (self-absorption) planes that translated a triangle in itself consists of 6 elements: three turns on the corners 0, 2π / 3. and 4π / 3. Around the point O, as well as three symmetries with respect to three direct, on which the bisector of the triangle is lying (the latter are also its heights and medians).
On the described circle of an arbitrary triangle $ABC$ There are exactly three points such that their direct Simson touches the circumference of the Euler triangle $ABC$ , and these points form right triangle. The sides of this triangle are parallel to the sides of the Frores triangle.
The equilateral triangle is both an equilibious triangle, that is, it is equal to all internal angles.
The equilateral triangle is a special case of an equifiable triangle, namely: twice an equally chaled triangle.

Theorems on the equilateral triangle or containing it

Direct Simson one of the properties

By number of parties

Right

Triangles

Quadrigons

see also



Polygons

Stars of polygons

Parquets on the plane

Right polyhedra and spherical parquets

Kepler polyhedra - Puenau

Honeycomb

When the next day, the sovereign said from him the officers: "You saved not one Russia; You saved Europe, "everything was then understood that the war was not over. One Kutuzov did not want to understand this and openly spoke his opinion that the new war could not improve the situation and increase the glory of Russia, but only may worsen its position and reduce the highest degree of fame, on which, in his opinion, Russia now stood Russia. He tried to prove the sovereign impossibility of tying new troops; He spoke about the serious position of populations, about the possibility of failures, etc. With such a mood, Feldmarshal, naturally, was only an interference and a brake of the upcoming war. To avoid collisions with an old man, I had a way out, consisting in the fact that, as in Austerlitz, and how at the beginning of a campaign at Barclay, remove it from under the commander-in-chief, not anxious, not announcing him about that the soil of power on which he stood , and transfer it to the sovereign itself. With this purpose, the headquarters was minted, and the whole essential strength of the Kutuzov headquarters was destroyed and moved to the sovereign. Tol, Konovnitsyn, Yermolov - received other appointments. Everyone said loudly that Field Marshal became very weak and upset his health. He had to be weak health, in order to convey his place to someone who intercepted him. And indeed, his health was weak. As naturally, and simply, and gradually came the Kutuzov from Turkey to the state chamber of St. Petersburg to collect the militia and then into the army, precisely when he was needed, just as naturally, gradually and just now, when the role of Kutuzov was played, to his place The new, demanded figure. The war of 1812 years of year, besides his dear to the Russian heart of the crowd, should have another - European. Behind the movement of peoples from the west to the East, the movement of peoples from the east to the West was to follow, and for this new war, a new figure was needed, having others than Kutuzov, properties, views movable by other motivations. Alexander is the first for the movement of peoples from the east to the West and to restore the borders of the peoples was as needed as Kutuzov was needed for salvation and glory of Russia. Kutuzov did not understand what Europe meant, equilibrium, Napoleon. He could not understand this. A representative of the Russian people, after the enemy was destroyed, Russia is released and put on the highest degree of fame, a Russian man, like Russian, there was nothing more. The representative of the People's War did not remain anything except death. And he died. Pierre, as it is the bigger part of the part, felt the whole severity of physical deprivation and stresses tested in captivity, only when these voltages and deprivation ran out. After his liberation from captivity, he came to the eagle and on the third day of his arrival, while he gathered in Kiev, fell ill and knew the patients in Orel for three months; It was done with him, as the Doctors said, bile hot. Despite the fact that the doctors treated him, they dreamed of blood and allowed to drink medicines, he all had recovered.

In the school year of geometry, a huge amount of time is paid to the study of triangles. Pupils calculate the angles, build biscomers and heights, find out what figures differ from each other, and how is the easiest way to find their area and perimeter. It seems that this is not useful in life, but sometimes it is still useful to learn, for example, how to determine that the triangle is equilateral or stupid. How to do it?

Types of triangles

Three points that do not lie on one straight line and segments that connect them. It seems that this figure is the simplest. What could be triangles if they have only three sides? In fact, options are quite a large number, and some of them pay special attention within the framework of the school year of geometry. The correct triangle is equilateral, that is, all its corners and the parties are equal. It has a number of remarkable properties, which will be discussed on.

It is equally equal to only two sides, and it is also quite interesting. At rectangular and how easy it is to guess, respectively, one of the corners are direct or stupid. At the same time, they can also be delicious.

There is a special called Egyptian. Its parties are 3, 4 and 5 units. In this case, it is rectangular. It is believed to be actively used by Egyptian land surveyors and architects for the construction of direct corners. It is believed that with his help the famous pyramids were erected.

Still, all the vertices of the triangle can lie on one straight line. In this case, it will be called degenerate, while everyone else is nondegenerate. They are one of the items of studying geometry.

Triangle equilateral

Of course, the correct figures are always the greatest interest. They seem more perfect, more elegant. Formulas for calculating their characteristics are often easier and shorter than for ordinary figures. This also applies to triangles. It is not surprising that when studying the geometry, they are paid quite a lot of attention: schoolchildren learn to distinguish the right figures from the rest, and also talk about some of their interesting characteristics.

Signs and properties

As it is easy to guess from the name, each side of the equilateral triangle is equal to two others. In addition, it has a number of signs, thanks to which you can define whether the figure is correct or not.

If at least one of the above signs is observed, then the triangle is equilateral. For proper figures, all the mentioned statements are fair.

All triangles have a number of remarkable properties. First, the middle line, that is, a segment dividing two sides in half and parallel to the third is half the base. Secondly, the sum of all angles of this figure is always equal to 180 degrees. In addition, in triangles there is another curious relationship. So, against most of the parties a larger angle and vice versa. But this, of course, does not have a relation to an equilateral triangle, because he has all the corners equal.

Inscribed and described circles

Often, in the course of geometry, students are also learning how the figures can interact with each other. In particular, circles inscribed in polygons or described near them are studied. What is it about?

In addition to such a circle, for which all sides of the polygon are tangent. Described - the one that has points of contact with all angles. For each triangle you can always build both the first and second circle, but only one of each type. Evidence of two of these

theorems are given in the school course of geometry.

In addition to calculating the parameters of the triangles themselves, some tasks also imply the calculation of the radii of these circles. And formulas in relation to
the equilateral triangle look like this:

where R is the radius of the inscribed circle, R is the radius of the circle described, and the length of the side of the triangle.

Calculation of height, perimeter and square

The main parameters, the calculation of which students are engaged in the study of geometry, remain unchanged for almost any shapes. It is a perimeter, an area and height. For ease of calculations, there are various formulas.

So, perimeter, that is, the length of all sides is calculated in the following ways:

P \u003d 3A \u003d 3√ ̅3R \u003d 6√ ̅3R, where A is the side of the correct triangle, R is the radius of the described circle, R is inscribed.

h \u003d (√ ̅3 / 2) * A, where A is the length of the parties.

Finally, the formula is derived from the standard, that is, the works of half the base on its height.

S \u003d (√ ̅3 / 4) * a 2, where a is the length of the parties.

Also, this value can be calculated through the parameters of the described or inscribed circle. For this, there are also special formulas:

S \u003d 3√ ̅3R 2 \u003d (3√ ̅3 / 4) * R 2, where R and R are, respectively, the radii of the inscribed and described circles.

Building

Another interesting type of tasks regarding the triangles is associated with the need to draw one or another figure using the minimum set.

tools: Circle and line without divisions.

In order to build a right triangle using only these devices, you must perform several steps.

It is necessary to draw a circle with any radius and with the center in an arbitrarily taken point A. It must be noted.
Next you need to spend directly through this point.
Crossing the circle and direct must be designated as in and C. All construction should be carried out with the highest possible accuracy.
Next, it is necessary to build another circumference with the same radius and center at a point with or arc with the corresponding parameters. The intersection points will be indicated as D and F.
Points B, F, D must be connected by segments. Equilance triangle built.

The solution of such tasks is usually a problem for schoolchildren, but this skill can be useful in everyday life.

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