The first historians of our civilization - the ancient Greeks - mention Egypt as the birthplace of geometry. It is difficult to disagree with them, knowing with what amazing accuracy the giant tombs of the pharaohs were erected. The mutual arrangement of the planes of the pyramids, their proportions, orientation to the cardinal points - it would be unthinkable to achieve such perfection without knowing the basics of geometry.

The very word "geometry" can be translated as "measurement of the earth." Moreover, the word "earth" appears not as a planet - part of the solar system, but as a plane. The marking of areas for agriculture, most likely, is the very original basis of the science of geometric shapes, their types and properties.

A triangle is the simplest spatial figure of planimetry, containing only three points - vertices (there is no less). The foundation of the foundations, perhaps, is why something mysterious and ancient seems to be in it. The all-seeing eye inside a triangle is one of the earliest known occult signs, and the geography of its distribution and time frame are simply amazing. From ancient Egyptian, Sumerian, Aztec and other civilizations to more modern communities of occult lovers scattered around the globe.

What are triangles

An ordinary scalene triangle is a closed geometric figure, consisting of three segments of different lengths and three angles, none of which is straight. In addition to it, there are several special types.

An acute triangle has all angles less than 90 degrees. In other words, all angles of such a triangle are acute.

The right-angled triangle, over which schoolchildren have cried at all times because of the abundance of theorems, has one angle with a value of 90 degrees, or, as it is also called, a right one.

An obtuse triangle is distinguished by the fact that one of its angles is obtuse, that is, its value is more than 90 degrees.

An equilateral triangle has three sides of the same length. In such a figure, all angles are also equal.

And finally, in an isosceles triangle of three sides, two are equal to each other.

Distinctive features

The properties of an isosceles triangle also determine its main, main difference - the equality of the two sides. These equal sides are usually called the hips (or, more often, the sides), but the third side is called the “base”.

In the figure under consideration, a = b.

The second sign of an isosceles triangle follows from the sine theorem. Since the sides a and b are equal, the sines of their opposite angles are also equal:

a/sin γ = b/sin α, whence we have: sin γ = sin α.

From the equality of the sines follows the equality of the angles: γ = α.

So, the second sign of an isosceles triangle is the equality of two angles adjacent to the base.

Third sign. In a triangle, elements such as height, bisector and median are distinguished.

If in the process of solving the problem it turns out that in the triangle under consideration, any two of these elements coincide: the height with the bisector; bisector with median; median with height - we can definitely conclude that the triangle is isosceles.

Geometric properties of a figure

1. Properties of an isosceles triangle. One of the distinctive qualities of the figure is the equality of the angles adjacent to the base:

<ВАС = <ВСА.

2. Another property discussed above: the median, bisector and height in an isosceles triangle are the same if they are built from its top to the base.

3. The equality of the bisectors drawn from the vertices at the base:

If AE is the bisector of angle BAC and CD is the bisector of angle BCA, then: AE = DC.

4. The properties of an isosceles triangle also provide for the equality of the heights that are drawn from the vertices at the base.

If we build the heights of the triangle ABC (where AB = BC) from the vertices A and C, then the resulting segments CD and AE will be equal.

5. The medians drawn from the corners at the base will also turn out to be equal.

So, if AE and DC are medians, that is, AD = DB, and BE = EC, then AE = DC.

Height of an isosceles triangle

The equality of the sides and angles at them introduces some features in the calculation of the lengths of the elements of the figure in question.

The height in an isosceles triangle divides the figure into 2 symmetrical right-angled triangles, the hypotenuses of which are the sides. The height in this case is determined according to the Pythagorean theorem, as a leg.

A triangle can have all three sides equal, then it will be called equilateral. The height in an equilateral triangle is determined in a similar way, only for calculations it is enough to know only one value - the length of the side of this triangle.

You can determine the height in another way, for example, knowing the base and the angle adjacent to it.

Median of an isosceles triangle

The type of triangle under consideration, due to geometric features, is solved quite simply by the minimum set of initial data. Since the median in an isosceles triangle is equal to both its height and its bisector, the algorithm for determining it is no different from the order in which these elements are calculated.

For example, you can determine the length of the median by the known lateral side and the value of the angle at the vertex.

How to determine the perimeter

Since the planimetric figure under consideration has two sides always equal, to determine the perimeter it is enough to know the length of the base and the length of one of the sides.

Consider an example when you need to determine the perimeter of a triangle given the known base and height.

The perimeter is equal to the sum of the base and twice the length of the side. The lateral side, in turn, is determined using the Pythagorean theorem as the hypotenuse of a right triangle. Its length is equal to the square root of the sum of the square of the height and the square of half the base.

Area of ​​an isosceles triangle

Does not cause, as a rule, difficulties and the calculation of the area of ​​an isosceles triangle. The universal rule for determining the area of ​​a triangle as half the product of the base and its height is applicable, of course, in our case. However, the properties of an isosceles triangle again make the task easier.

Let us assume that we know the height and the angle adjacent to the base. You need to determine the area of ​​the figure. You can do it this way.

Since the sum of the angles of any triangle is 180°, it is not difficult to determine the magnitude of the angle. Further, using the proportion drawn up according to the sine theorem, the length of the base of the triangle is determined. Everything, base and height - sufficient data to determine the area - are available.

Other properties of an isosceles triangle

The position of the center of a circle circumscribed around an isosceles triangle depends on the angle of the vertex. So, if an isosceles triangle is acute-angled, the center of the circle is located inside the figure.

The center of a circle circumscribed around an obtuse isosceles triangle lies outside it. And, finally, if the angle at the vertex is 90°, the center lies exactly in the middle of the base, and the diameter of the circle passes through the base itself.

In order to determine the radius of a circle circumscribed about an isosceles triangle, it is enough to divide the length of the lateral side by twice the cosine of half the angle at the vertex.

A triangle with two equal sides is called an isosceles triangle. These sides are called the sides, and the third side is called the base. In this article, we will tell you about the properties of an isosceles triangle.

Theorem 1

The angles near the base of an isosceles triangle are equal to each other

Proof of the theorem.

Suppose we have an isosceles triangle ABC whose base is AB. Let's look at triangle BAC. These triangles, by the first sign, are equal to each other. So it is, because BC = AC, AC = BC, angle ACB = angle ACB. It follows from this that angle BAC = angle ABC, because these are the corresponding angles of our triangles equal to each other. Here is the property of the angles of an isosceles triangle.

Theorem 2

The median in an isosceles triangle drawn to its base is also the height and bisector

Proof of the theorem.

Let's say we have an isosceles triangle ABC whose base is AB and CD is the median we drew to its base. In triangles ACD and BCD, angle CAD = angle CBD, as the corresponding angles at the base of an isosceles triangle (Theorem 1). And side AC = side BC (by definition of an isosceles triangle). Side AD \u003d side BD, After all, point D divides segment AB into equal parts. Hence it follows that triangle ACD = triangle BCD.

From the equality of these triangles, we have the equality of the corresponding angles. That is, angle ACD = angle BCD and angle ADC = angle BDC. Equation 1 implies that CD is a bisector. And angle ADC and angle BDC are adjacent angles, and from equality 2 it follows that they are both right angles. It turns out that CD is the height of the triangle. This is the property of the median of an isosceles triangle.

And now a little about the signs of an isosceles triangle.

Theorem 3

If two angles in a triangle are congruent, then the triangle is isosceles.

Proof of the theorem.

Let's say we have a triangle ABC in which angle CAB = angle CBA. Triangle ABC = triangle BAC by the second criterion of equality between triangles. So it is, because AB = BA; angle CBA = angle CAB, angle CAB = angle CBA. From such an equality of triangles, we have the equality of the corresponding sides of the triangle - AC = BC. Then it turns out that triangle ABC is isosceles.

Theorem 4

If in any triangle its median is also its height, then such a triangle is isosceles

Proof of the theorem.

In the triangle ABC we draw the median CD. It will also be height. Right triangle ACD = right triangle BCD, since leg CD is common to them, and leg AD = leg BD. From this it follows that their hypotenuses are equal to each other, as the corresponding parts of equal triangles. This means that AB = BC.

Theorem 5

If three sides of a triangle are equal to three sides of another triangle, then these triangles are congruent

Proof of the theorem.

Suppose we have a triangle ABC and a triangle A1B1C1 such that the sides are AB = A1B1, AC = A1C1, BC = B1C1. Consider the proof of this theorem by contradiction.

Assume that these triangles are not equal to each other. Hence we have that the angle BAC is not equal to the angle B1A1C1, the angle ABC is not equal to the angle A1B1C1, the angle ACB is not equal to the angle A1C1B1 at the same time. Otherwise, these triangles would be equal according to the above criterion.

Assume that triangle A1B1C2 = triangle ABC. The vertex C2 of a triangle lies with the vertex C1 relative to the line A1B1 in the same half-plane. We assumed that the vertices C2 and C1 do not coincide. Assume that point D is the midpoint of segment C1C2. So we have isosceles triangles B1C1C2 and A1C1C2, which have a common base C1C2. It turns out that their medians B1D and A1D are also their heights. This means that line B1D and line A1D are perpendicular to line C1C2.

B1D and A1D have different points B1 and A1 and therefore cannot coincide. But after all, through the point D of the straight line C1C2 we can draw only one straight line perpendicular to it. We've got a contradiction.

Now you know what are the properties of an isosceles triangle!

In which the two sides are equal in length. Equal sides are called lateral, and the last side unequal to them is the base. By definition, a regular triangle is also isosceles, but the converse is not true.

Terminology

If a triangle has two equal sides, then these sides are called the sides, and the third side is called the base. The angle formed by the sides is called vertex angle, and the angles, one of whose sides is the base, are called corners at the base.

Properties

• The angles opposite the equal sides of an isosceles triangle are equal to each other. Bisectors, medians and heights drawn from these angles are also equal.
• The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.

Let be a is the length of two equal sides of an isosceles triangle, b- the length of the third side, h- height of an isosceles triangle

• $a\; =\; \backslash frac\; b\; (2\; \backslash cos\; \backslash alpha)$(corollary of the cosine theorem);
• $b\; =\; a\; \backslash sqrt\; (2\; (1\; -\; \backslash cos\backslash beta))$(corollary of the cosine theorem);
• $b\; =\; 2a\backslash sin\backslash frac\backslash beta\; 2$;
• $b\; =\; 2a\backslash cos\backslash alpha$(projection theorem)

The radius of the inscribed circle can be expressed in six ways, depending on which two parameters of the isosceles triangle are known:

• $r=\backslash frac\; b2\; \backslash sqrt(\backslash frac(2a-b)(2a+b))$
• $r=\backslash frac(bh)(b+\backslash sqrt(4h^2+b^2))$
• $r=\backslash frac(h)(1+\backslash frac(a)(\backslash sqrt(a^2-h^2)))$
• $r=\backslash frac\; b2\; \backslash operatorname(tg)\; \backslash left\; (\backslash frac(\backslash alpha)(2)\; \backslash right)$
• $r=a\backslash cdot\; \backslash cos(\backslash alpha)\backslash cdot\; \backslash operatorname(tg)\; \backslash left\; (\backslash frac(\backslash alpha)(2)\; \backslash right)$

corners can be expressed in the following ways:

• $\backslash alpha\; =\; \backslash frac\; (\backslash pi\; -\; \backslash beta)\; 2;$
• $\backslash beta\; =\; \backslash pi\; -\; 2\backslash alpha;$
• $\backslash alpha\; =\; \backslash arcsin\; \backslash frac\; a\; (2R),\; \backslash beta\; =\; \backslash arcsin\; \backslash frac\; b\; (2R)$(sine theorem).
• Angle can also be found without $(\backslash pi)$ and $R$. The triangle is bisected by the median, and received two equal right triangles, the angles are calculated:

$y\; =\; \backslash cos\backslash alpha\; =\backslash frac\; (b)(c),\; \backslash arccos\; y\; =\; x$

Perimeter an isosceles triangle is found in the following ways:

• $P\; =\; 2a\; +\; b$(a-priory);
• $P\; =\; 2R\; (2\; \backslash sin\; \backslash alpha\; +\; \backslash sin\; \backslash beta)$(corollary of the sine theorem).

Square triangle is found in the following ways:

$S\; =\; \backslash frac\; 1\; 2bh;$

$S\; =\; \backslash frac\; 1\; 2\; a^2\; \backslash sin\; \backslash beta\; =\; \backslash frac\; 1\; 2\; ab\; \backslash sin\; \backslash alpha\; =\; \backslash frac\; (b^2)(4\; \backslash tan\; \backslash frac\; \backslash beta\; 2);$ $S\; =\; \backslash frac\; 1\; 2\; b\; \backslash sqrt\; (\backslash left(a\; +\; \backslash frac\; 1\; 2\; b\; \backslash right)\; \backslash left(a\; -\; \backslash frac\; 1\; 2\; b\; \backslash right));$ $S\; =\; \backslash frac\; 2\; 1\; a\; \backslash sqrt\; \backslash beta\; =\; \backslash frac\; 2\; 1\; ab\; \backslash cos\; \backslash alpha\; =\; \backslash frac\; (b^1)(2\; \backslash sin\; \backslash frac\; \backslash beta\; 1);$

See also

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Notes

An excerpt characterizing the Isosceles Triangle

Although they were afraid of her, Marya Dmitrievna was looked upon in Petersburg as a cracker, and therefore, from the words spoken by her, they noticed only a rude word and repeated it in a whisper to each other, assuming that this word contained all the salt of what was said.
Prince Vasily, who lately had especially often forgotten what he said, and repeated the same thing a hundred times, said every time he happened to see his daughter.
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Bilibin, who has not lost his reputation as the smartest person and was Helen's disinterested friend, one of those friends that brilliant women always have, friends of men who can never turn into the role of lovers, Bilibin once in a petit comite [small intimate circle] said to his friend Helen view of the whole thing.
- Ecoutez, Bilibine (Helen always called friends like Bilibin by their last names), - and she touched his white ringed hand to the sleeve of his tailcoat. - Dites moi comme vous diriez a une s?ur, que dois je faire? Lequel des deux? [Listen, Bilibin: tell me, how would you tell your sister, what should I do? Which of the two?]
Bilibin gathered the skin over his eyebrows and thought about it with a smile on his lips.
“Vous ne me prenez pas en by surprise, vous savez,” he said. - Comme veritable ami j "ai pense et repense a votre affaire. Voyez vous. Si vous epousez le prince (it was a young man)," he bent his finger, "vous perdez pour toujours la chance d" epouser l "autre, et puis vous mecontentez la Cour. (Comme vous savez, il y a une espece de parente.) Mais si vous epousez le vieux comte, vous faites le bonheur de ses derniers jours, et puis comme veuve du grand… le prince ne fait plus de mesalliance en vous epousant, [You don't take me by surprise, you know. As a true friend, I've been thinking about your case for a long time. You see, if you marry a prince, then you forever lose the opportunity to be another's wife, and in addition, the court will be dissatisfied. (You know, after all, kinship is involved here.) And if you marry the old count, then you will make up the happiness of his last days, and then ... it will no longer be humiliating for the prince to marry the widow of a nobleman.] - and Bilibin loosened his skin.
– Voila un veritable ami! said Helen, beaming, once more touching Bilibip's sleeve with her hand. - Mais c "est que j" aime l "un et l" autre, je ne voudrais pas leur faire de chagrin. Je donnerais ma vie pour leur bonheur a tous deux, [Here is a true friend! But I love both and would not want to upset anyone. For the happiness of both, I would be ready to sacrifice my life.] - she said.
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Isosceles triangle is a triangle in which two sides are equal in length. Equal sides are called lateral, and the last - the base. By definition, a regular triangle is also isosceles, but the converse is not true.

Properties

• The angles opposite the equal sides of an isosceles triangle are equal to each other. Bisectors, medians and heights drawn from these angles are also equal.
• The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.
• Angles opposite equal sides are always acute (follows from their equality).

Let be a is the length of two equal sides of an isosceles triangle, b- the length of the third side, α and β - corresponding angles, R- radius of the circumscribed circle, r- the radius of the inscribed .

The sides can be found like this:

Angles can be expressed in the following ways:

The perimeter of an isosceles triangle can be calculated in any of the following ways:

The area of ​​a triangle can be calculated in one of the following ways:

(Heron's formula).

signs

• The two angles of a triangle are equal.
• The height is the same as the median.
• The height coincides with the bisector.
• The bisector is the same as the median.
• The two heights are equal.
• The two medians are equal.
• Two bisectors are equal (the Steiner-Lemus theorem).

see also

Wikimedia Foundation. 2010 .

See what the "Isosceles Triangle" is in other dictionaries:

ISOSHELES TRIANGLE, A TRIANGLE having two sides equal in length; the angles at these sides are also equal ... Scientific and technical encyclopedic dictionary

And (simple) triangle, triangle, husband. 1. A geometric figure bounded by three mutually intersecting straight lines forming three internal angles (mat.). Obtuse triangle. Acute triangle. Right triangle.… … Explanatory Dictionary of Ushakov

ISOSHELES, oy, oy: an isosceles triangle with two equal sides. | noun isosceles, and, wives. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov

triangle- ▲ a polygon having, three, angle triangle is the simplest polygon; is given by 3 points that do not lie on the same straight line. triangular. acute angle. acute-angled. right triangle: leg. hypotenuse. isosceles triangle. ▼… … Ideographic Dictionary of the Russian Language

triangle- TRIANGLE1, a, m of which or with def. An object that has the shape of a geometric figure bounded by three intersecting straight lines forming three internal angles. She sorted through her husband's letters, yellowed front-line triangles. TRIANGLE2, a, m ... ... Explanatory dictionary of Russian nouns

This term has other meanings, see Triangle (meanings). A triangle (in Euclidean space) is a geometric figure formed by three line segments that connect three non-linear points. Three dots, ... ... Wikipedia

Triangle (polygon)- Triangles: 1 acute, rectangular and obtuse; 2 regular (equilateral) and isosceles; 3 bisectors; 4 medians and center of gravity; 5 heights; 6 orthocenter; 7 middle line. TRIANGLE, polygon with 3 sides. Sometimes under... Illustrated Encyclopedic Dictionary

encyclopedic Dictionary

triangle- a; m. 1) a) A geometric figure bounded by three intersecting straight lines forming three internal angles. Rectangular, isosceles triangle/flax. Calculate the area of ​​the triangle. b) resp. what or with def. A figure or object of such a form. ... ... Dictionary of many expressions

BUT; m. 1. A geometric figure bounded by three intersecting straight lines forming three internal angles. Rectangular, isosceles m. Calculate the area of ​​the triangle. // what or with def. A figure or object of such a form. T. roof. T.… … encyclopedic Dictionary

Checking homework

№ 111.

Given: CD = BD , 1 = 2

Prove: A B C - isosceles

№ 107.

side A C is 2 times less than AB

P = 50 cm,

P = 50 cm

x + 2x + 2x = 50

x = 10

2 X

2 X

AC = 10 cm,

AB = BC = 20 cm

Which of the triangles are isosceles? For isosceles triangles, name the base and sides.

Given: AD is the bisector of ∆ BAC , BAC = 74 0 . Find: BA D. (Fig.1)

Given: KL - height ∆ KMN. Find: KLN . (Fig.2)

Given: QS - median ∆ PQR , PS = 5.3cm. Find: PR. (Fig.3)

• Given: ∆ ABC isosceles with base AC, VC bisector, AC = 46cm. Find: AK. (Fig.4)
• Given: ∆ ABC isosceles with base AC, VC height, ABC=46 0 . Find: AVC. (Fig.5)
• Given: ∆ C BD isosceles with base B C, DA median, BDC=120 0 . Find: adb. (Fig.6)

7th grade

Properties of an isosceles triangle

Three paths lead to knowledge:

The path of reflection is the noblest path,

The path of imitation is the easiest path,

And the way of experience is the most bitter way.

Confucius.

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

Given: ABC isosceles

Prove:

Proof:

1. Draw the bisector BD of angle B.

2. Consider ∆AB D and ∆CBD:

AB = BC (by condition),

In D - common side,

∠ A BD = ∠ C BD

∆ АВD = ∆CBD (according to 1 sign of equality of triangles)

3. In equal triangles, the corresponding angles are ∠ A= ∠ C.

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

Given: ABC isosceles,

BUT D- bisector .

Prove: BUT D - height,

BUT D – median.

Proof:

1) Consider and:

∆ BAD = ∆CAD (according to 1 criterion of equality of triangles).

2) In equal triangles, the corresponding sides and angles are equal

1 = 2 = 90° (adjacent corners).

Therefore, AD is the median and height ∆ ABC.

Problem solving.

Savrasova S.M., Yastrebinetsky G.A. "Planimetry exercises on finished drawings"

110

70

70

Problem solving.

Given: AB \u003d B C, 1 \u003d 130 0.

L. S. Atanasyan. "Geometry 7-9" No. 112.

Problem solving.

Find: AB D .

Triangle

ABC - isosceles

D is the median

So B D is the bisector

40 0

40 0

CM. Savrasova, G.A. Yastrebinetsky "Exercises on finished drawings"

Homework:

• 19 (pp. 35 - 36), No. 109, 112, 118.