Two equal sides are called lateral, the third is called the base.
Definition 8. If all three sides of a triangle are equal, then the triangle is called equilateral.
He is a private species isosceles triangle.
Theorem 18. The height of an isosceles triangle, lowered to the base, is at the same time the bisector of the angle between equal sides, the median and the axis of symmetry of the base.
Proof. Let us lower the height to the base of the isosceles triangle. She will divide it into two equal parts (along the leg and hypotenuse) right triangle. Angles A and C are equal, and the height also divides the base in half and will be the axis of symmetry of the entire figure under consideration.
This theorem can also be formulated as follows:
Theorem 18.1. The median of an isosceles triangle, lowered to the base, is also the bisector of the angle between equal sides, the height and the axis of symmetry of the base.
Theorem 18.2. The bisector of an isosceles triangle, lowered to the base, is simultaneously the height, median and axis of symmetry of the base.
Theorem 18.3. The axis of symmetry of an isosceles triangle is simultaneously the bisector of the angle between equal sides, the median and the altitude.
The proof of these consequences also follows from the equality of the triangles into which an isosceles triangle is divided.
Theorem 19.
The angles at the base of an isosceles triangle are equal.
Proof. Let us lower the height to the base of the isosceles triangle. It will divide it into two equal (along the leg and hypotenuse) right triangles, which means the corresponding angles are equal, i.e. ∠ A=∠ C
The criteria for an isosceles triangle come from Theorem 1 and its corollaries and Theorem 2.
Theorem 20.
If two of the indicated four lines (height, median, bisector, axis of symmetry) coincide, then the triangle will be isosceles (which means all four lines will coincide).
Theorem 21.
If any two angles of a triangle are equal, then it is isosceles.
Proof: Similar to the proof of the direct theorem, but using the second criterion for the equality of triangles. The center of gravity, the centers of the circumcircle and incircle, and the point of intersection of the altitudes of an isosceles triangle all lie on its axis of symmetry, i.e. on high.
An equilateral triangle is isosceles for each pair of its sides. Due to the equality of all its sides, all three angles of such a triangle are equal. Considering that the sum of the angles of any triangle is equal to two right angles, we see that each of the angles of an equilateral triangle is equal to 60°. Conversely, to ensure that all sides of a triangle are equal, it is enough to check that two of its three angles are equal to 60°.
Theorem 22
. In an equilateral triangle, all the remarkable points coincide: the center of gravity, the centers of the inscribed and circumscribed circles, the point of intersection of the altitudes (called the orthocenter of the triangle).
Theorem 23
. If two of the indicated four points coincide, then the triangle will be equilateral and, as a consequence, all four named points will coincide.
Indeed, such a triangle will turn out, according to the previous one, isosceles with respect to any pair of sides, i.e. equilateral. An equilateral triangle is also called a regular triangle. 
The area of an isosceles triangle is equal to half the product of the square of the side side and the sine of the angle between the sides
Consider this formula for an equilateral triangle, then the alpha angle will be equal to 60 degrees. Then the formula will change to this:
Theorem d1
Proof:. In an isosceles triangle, the medians drawn to the sides are equal.
Let ABC be an isosceles triangle (AC = BC), AK and BL its medians. Then triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have common side AB, sides AL and BK are equal as halves of the lateral sides of an isosceles triangle, and angles LAB and KBA are equal as the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB are equal. But AK and LB are the medians of an isosceles triangle drawn to its lateral sides.
Theorem d2
Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its bisectors. Triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have a common side AB, angles LAB and KBA are equal as the base angles of an isosceles triangle, and angles LBA and KAB are equal as half the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB - the bisectors of triangle ABC - are congruent. The theorem is proven.
Theorem d3
. In an isosceles triangle, the heights lowered to the sides are equal.
Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its altitudes. Then angles ABL and KAB are equal, since angles ALB and AKB are right angles, and angles LAB and ABK are equal as the base angles of an isosceles triangle. Consequently, triangles ALB and AKB are equal according to the second criterion for the equality of triangles: they have a common side AB, angles KAB and LBA are equal according to the above, and angles LAB and KBA are equal as the base angles of an isosceles triangle. If the triangles are congruent, their sides AK and BL are also congruent. Q.E.D.
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IN school course geometry great amount time is devoted to studying 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? There are actually quite a few options. a large number of, 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. Firstly, middle line, that is, a 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 minimum set
tools: compass and ruler without divisions.
In order to construct a regular triangle using only these devices, you need to follow several steps.
- You need to draw a circle with any radius and with a center at an arbitrary point A. It must be marked.
- Next you need to draw a straight line through this point.
- 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.
- Next, you need to construct 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.
- 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.
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