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Definition 7. Any triangle whose two sides are equal is called isosceles.Two equal sides are called lateral, the third is called the base.
Definition 8. If all three sides of a triangle are equal, then such a triangle is called equilateral.
He is a private species isosceles triangle.
Theorem 18. The height of an isosceles triangle, lowered to the base, is also the bisector of the angle between equal sides, the median and the axis of symmetry of the base.
Proof. Let us drop the height to the base of the isosceles triangle. She will divide it into two equal (along the leg and hypotenuse) right-angled triangles. 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 in question.
Also, this theorem can 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 at the same time 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, median and height.
The proof of these corollaries also follows from the equality of the triangles into which the isosceles triangle is divided.
Theorem 19.
The angles at the base of an isosceles triangle are equal.
Proof. Let us drop the height to the base of the isosceles triangle. She will divide it into two equal (in leg and hypotenuse) right triangles, which means corresponding angles are equal, i.e. ∠ A = ∠ C
The criteria for an isosceles triangle follow from Theorem 1 and its corollaries and Theorem 2.
Theorem 20.
If two of these four lines (height, median, bisector, axis of symmetry) coincide, then the triangle will be isosceles (which means that all four lines will also 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 test for the equality of triangles. The center of gravity, the centers of the circumscribed and inscribed circles and the point of intersection of the heights 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. In view of 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 60 °. Conversely, to make sure 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 heights (called the orthocenter of the triangle).
Theorem 23
... If two of these four points coincide, then the triangle will be equilateral and, as a result, all four named points will coincide.
Indeed, such a triangle will, according to the previous one, be 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 lateral side and the sine of the angle between the sides 
Consider this formula for an equilateral triangle, then the alpha angle will be 60 degrees. Then the formula will change to look like this:
Theorem d1
... In an isosceles triangle, the medians drawn to the lateral sides are equal.
Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its medians. Then triangles AKB and ALB are equal according to the second sign of equality of triangles. They have a common side AB, sides AL and BK are equal as halves of the lateral sides of an isosceles triangle, and the angles LAB and KBA are equal as angles at the base of an isosceles triangle. Since the triangles are equal, 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
... In an isosceles triangle, the bisectors drawn to the sides are equal.
Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its bisectors. Triangles AKB and ALB are equal in the second sign of triangle equality. They have a common side AB, the angles LAB and KBA are equal as the angles at the base of an isosceles triangle, and the angles LBA and KAB are equal as half the angles at the base of an isosceles triangle. Since the triangles are equal, their sides AK and LB - the bisectors of triangle ABC - are equal. The theorem is proved.
Theorem d3
... In an isosceles triangle, the heights dropped to the lateral sides are equal.
Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its heights. Then the angles ABL and KAB are equal, since the angles ALB and AKB are right, and the angles LAB and ABK are equal as the angles at the base of an isosceles triangle. Therefore, triangles ALB and AKB are equal according to the second sign of triangle equality: they have a common side AB, the angles KAB and LBA are equal according to the above, and the angles LAB and KBA are equal as the angles at the base of an isosceles triangle. If the triangles are equal, their sides AK and BL are also equal. Q.E.D.
V school course geometry great amount time is devoted to the study of triangles. Pupils calculate angles, build bisectors and heights, find out how the figures differ from each other, and the easiest way to find their area and perimeter. It seems that this will not come in handy in life, but sometimes it is still useful to learn, for example, how to determine that a triangle is equilateral or obtuse. How can this be done?
Types of triangles
Three points that do not lie on one straight line, and the line segments that connect them. It seems that this figure is the simplest one. What can be triangles if they have only three sides? In fact, there are quite a few options a large number of, and some of them are given Special attention as part of 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 below.
The isosceles have only two sides equal, and they are also quite interesting. In a rectangular one, and as you might guess, one of the corners is straight or obtuse, respectively. However, they can also be isosceles.
There is also a special one called Egyptian. Its sides are equal to 3, 4 and 5 units. Moreover, it is rectangular. It is believed to have been actively used by Egyptian surveyors and architects to build right angles. It is believed that the famous pyramids were erected with his 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 are non-degenerate. It is they who are one of the subjects of the study of geometry.
Equilateral triangle
Of course, correct figures are always of the greatest interest. They seem to be more perfect, more graceful. Formulas for calculating their characteristics are often simpler and shorter than for ordinary shapes. This also applies to triangles. It is not surprising that a lot of attention is paid to them in the study of geometry: students are taught to distinguish the correct figures from the rest, and also talk 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 possesses a number of features, thanks to which it is possible to determine whether the figure is correct or not.
If at least one of the above signs is observed, then the triangle is equilateral. For correct figure all the above statements are true.
All triangles have a number of remarkable properties. First, the middle line, that is, the segment dividing the 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 180 degrees. In addition, there is another curious relationship in triangles. So, there is a larger angle opposite the larger side 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 explore how shapes can interact with each other. In particular, circles inscribed in or circumscribed about polygons are studied. What is it about?
An inscribed circle is a circle for which all sides of the polygon are tangent. Described - one that has points of contact with all corners. For each triangle, you can always build both the first and the second circle, 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 tasks also involve calculating the radii of these circles. And formulas applied to
equilateral triangle are as follows:
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.
Calculating Height, Perimeter, and Area
The main parameters, which are calculated by schoolchildren during the study of geometry, remain unchanged for almost any figure. These are the perimeter, area, and height. Various formulas exist for ease of calculation.
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 a regular triangle, R is the radius of the circumcircle, r is the circumcircle.
h = (√ ̅3 / 2) * a, where a is the side length.
Finally, the formula is derived from the standard, that is, the product of half the base by its height.
S = (√ ̅3 / 4) * a 2, where a is the side length.
Also, this value can be calculated through the parameters of the circumcircle 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.
Building
Another interesting type of problem, including triangles, is associated with the need to draw a particular shape using a minimal set
instruments: a compass and a ruler without divisions.
In order to build a regular triangle using only these devices, you need to follow several steps.
- It is necessary to draw a circle with any radius and with the center at an arbitrary point A. It must be noted.
- 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 build another circle with the same radius and center at point C or an arc with the appropriate parameters. The intersection points will be marked as D and F.
- Points B, F, D must be connected with segments. An equilateral triangle is built.
Solving such problems is usually a problem for schoolchildren, but this skill can be useful in everyday life.
