What are the values ​​of the elements in an equilateral triangle? Regular triangle

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Regular triangle R- radius of the circumscribed circle, r- radius of the inscribed circle.

• The radius of the inscribed circle of a regular triangle, expressed in terms of its side:

$r\; =\; \backslash frac(\backslash sqrt\; 3)(6)\; a$

• The radius of the circumscribed circle of a regular triangle, expressed in terms of its side:

$R\; =\; \backslash frac(\backslash sqrt\; 3)(3)\; a$

• Perimeter of a regular triangle:

$P\; =\; 3a\; =\; 3\; \backslash sqrt\; 3\; R\; =\; 6\; \backslash sqrt\; 3\; r$

• Altitudes, medians and bisectors of a regular triangle:

$h\; =\; m\; =\; l\; =\; \backslash frac(\backslash sqrt\; 3)(2)\; a$

• The area of ​​a regular triangle is calculated using the formulas:

$S\; =\; \backslash frac(\backslash sqrt\; 3)(4)\; a^2\; =\; \backslash frac(3\; \backslash sqrt\; 3)(4)\; R^2\; =\; 3\; \backslash sqrt\; 3\; r^2\; =\; \backslash frac(\backslash sqrt\; 3)(36)\; P\; ^2$

• The radius of the circumcircle is equal to twice the radius of the inscribed circle:

$R\; =\; 2r$

• You can tile a plane with regular triangles.

• In an equilateral triangle, the circle of nine points coincides with the incircle.

• For an equilateral triangle T, the group of movements (self-alignments) of the plane that transfer the triangle into itself consists of 6 elements: three rotations through angles 0, 2π ⁄ 3 And 4π ⁄ 3 around the point O, as well as three symmetries with respect to the three lines on which the bisectors of the triangle lie (the latter are also its altitudes and medians).
• On the circumcircle of an arbitrary triangle $ABC$ there are exactly three points such that their Simson line is tangent to the Euler circle of the triangle $ABC$, and these points form regular triangle. The sides of this triangle are parallel to the sides of the Morley triangle.
• An equilateral triangle is also an equiangular triangle, that is, all its interior angles are equal.
• An equilateral triangle is a special case of an isosceles triangle, namely a doubly isosceles triangle.

see also

Theorems about or containing an equilateral triangle

• Simson's straight line is one of the properties

By number of sides Correct Triangles Quadrilaterals see also

Polygons Star polygons Parquets on a plane Regular polyhedra and spherical parquets Kepler-Poinsot polyhedra Honeycomb When the next morning the sovereign said to the officers gathered at his place: “You saved more than just Russia; you saved Europe,” everyone already understood that the war was not over. Only Kutuzov did not want to understand this and openly expressed his opinion that a new war could not improve the situation and increase the glory of Russia, but could only worsen its position and reduce the highest degree of glory on which, in his opinion, Russia now stood. He tried to prove to the sovereign the impossibility of recruiting new troops; spoke about the difficult situation of the population, the possibility of failure, etc. In such a mood, the field marshal, naturally, seemed to be only a hindrance and a brake on the upcoming war. To avoid clashes with the old man, a way out was found by itself, which consisted in, as at Austerlitz and as at the beginning of the campaign under Barclay, to remove from under the commander-in-chief, without disturbing him, without announcing to him that the ground of power on which he stood , and transfer it to the sovereign himself. For this purpose, the headquarters was gradually reorganized, and all the significant strength of Kutuzov’s headquarters was destroyed and transferred to the sovereign. Tol, Konovnitsyn, Ermolov - received other appointments. Everyone said loudly that the field marshal had become very weak and was upset about his health. He had to be in poor health in order to transfer his place to the one who took his place. And indeed, his health was poor. Just as naturally, and simply, and gradually, Kutuzov came from Turkey to the treasury chamber of St. Petersburg to collect the militia and then into the army, precisely when he was needed, just as naturally, gradually and simply now, when Kutuzov’s role was played, to take his place a new, needed figure appeared. The war of 1812, in addition to its national significance dear to the Russian heart, should have had another – European one. The movement of peoples from West to East was to be followed by the movement of peoples from East to West, and for this new war a new figure was needed, with different properties and views than Kutuzov, driven by different motives. Alexander the First was as necessary for the movement of peoples from east to west and for the restoration of the borders of peoples as Kutuzov was necessary for the salvation and glory of Russia. Kutuzov did not understand what Europe, balance, Napoleon meant. He couldn't understand it. The representative of the Russian people, after the enemy was destroyed, Russia was liberated and placed at the highest level of its glory, the Russian person, as a Russian, had nothing more to do. The representative of the people's war had no choice but death. And he died. Pierre, as most often happens, felt the full weight of the physical deprivations and stresses experienced in captivity only when these stresses and deprivations ended. After his release from captivity, he came to Orel and on the third day of his arrival, while he was going to Kyiv, he fell ill and lay sick in Orel for three months; As the doctors said, he suffered from bilious fever. Despite the fact that the doctors treated him, bled him and gave him medicine to drink, he still recovered.

In the school geometry course, a huge amount of time is devoted to the study of triangles. Students calculate angles, construct bisectors and altitudes, find out how shapes differ from each other, and the easiest way to find their area and perimeter. It seems that this will not be useful in life, but sometimes it is still useful to learn, for example, how to determine whether a triangle is equilateral or obtuse. How to do this?

Types of Triangles

Three points that do not lie on the same line, and the segments that connect them. It seems that this figure is the simplest. What kind of triangles can they be if they only have three sides? In fact, there are quite a large number of options, and some of them are given special attention in the school geometry course. A regular triangle is equilateral, that is, all its angles and sides are equal. It has a number of remarkable properties, which will be discussed further.

An isosceles has only two equal sides, and is also quite interesting. In a rectangular one, as you might guess, one of the angles is straight or obtuse, respectively. Moreover, they can also be isosceles.

There is also a special one called Egyptian. Its sides are 3, 4 and 5 units. Moreover, it is rectangular. It is believed that it was actively used by Egyptian surveyors and architects to construct right angles. It is believed that the famous pyramids were built with its help.

And yet all the vertices of a triangle can lie on the same straight line. In this case, it will be called degenerate, while all the others will be called non-degenerate. They are one of the subjects of studying geometry.

Equilateral triangle

Of course, the correct figures always cause the greatest interest. They seem more perfect, more graceful. The formulas for calculating their characteristics are often simpler and shorter than for ordinary figures. This also applies to triangles. It is not surprising that when studying geometry they are given quite a lot of attention: schoolchildren are taught to distinguish the correct figures from the rest, and are also told about some of their interesting characteristics.

Signs and properties

As you might guess from the name, each side of an equilateral triangle is equal to the other two. In addition, it has a number of features that help you determine whether the figure is correct or not.

If at least one of the above signs is observed, then the triangle is equilateral. For the correct figure, all of the above statements are true.

All triangles have a number of remarkable properties. First, the middle line, that is, the segment dividing two sides in half and parallel to the third, is equal to half the base. Secondly, the sum of all the angles of this figure is always equal to 180 degrees. In addition, there is another interesting relationship in triangles. So, opposite the larger side lies the larger angle and vice versa. But this, of course, has nothing to do with an equilateral triangle, because all its angles are equal.

Inscribed and circumscribed circles

Often in a geometry course, students also learn how shapes can interact with each other. In particular, circles inscribed in polygons or described around them are studied. What is it about?

An inscribed circle is a circle for which all sides of the polygon are tangent. Described - the one that has points of contact with all corners. For each triangle, you can always construct both the first and second circles, but only one of each type. Evidence of these two

theorems are given in the school geometry course.

In addition to calculating the parameters of the triangles themselves, some problems also involve calculating the radii of these circles. And formulas for
equilateral triangle look like this:

where r is the radius of the inscribed circle, R is the radius of the circumscribed circle, a is the length of the side of the triangle.

Calculation of height, perimeter and area

The basic parameters that schoolchildren calculate while studying geometry remain unchanged for almost any figure. These are perimeter, area and height. To simplify calculations, there are various formulas.

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

P = 3a = 3√ ̅3R = 6√ ̅3r, where a is the side of an equilateral triangle, R is the radius of the circumscribed circle, r is the inscribed circle.

h = (√ ̅3/2)*a, where a is the length of the side.

Finally, the formula is derived from the standard one, that is, the product of half the base and its height.

S = (√ ̅3/4)*a 2, where a is the length of the side.

This value can also be calculated through the parameters of a circumscribed or inscribed circle. There are also special formulas for this:

S = 3√ ̅3r 2 = (3√ ̅3/4)*R 2, where r and R are the radii of the inscribed and circumscribed circles, respectively.

Construction

Another interesting type of problem, including triangles, involves the need to draw a particular figure using a minimal set

tools: compass and ruler without divisions.

In order to construct a regular triangle using only these devices, you need to follow several steps.

1. You need to draw a circle with any radius and with a center at an arbitrary point A. It must be marked.
2. Next you need to draw a straight line through this point.
3. The intersections of a circle and a straight line must be designated as B and C. All constructions must be carried out with the greatest possible accuracy.
4. Next, you need to build another circle with the same radius and center at point C or an arc with the appropriate parameters. The intersection points will be designated D and F.
5. Points B, F, D must be connected by segments. An equilateral triangle is constructed.

Solving such problems is usually a problem for schoolchildren, but this skill can be useful in everyday life.