Triangle) or pass outside the triangle at an obtuse triangle.

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✪ HEIGHT MEDIAN BIsectrix of a triangle Grade 7

✪ bisector, median, altitude of a triangle. Geometry 7th grade

✪ Grade 7, lesson 17, Medians, bisectors and altitudes of a triangle

✪ Median, bisector, altitude of triangle | Geometry

✪ How to find the length of the bisector, median and height? | Nerd with me #031 | Boris Trushin

Subtitles

Properties of the point of intersection of three altitudes of a triangle (orthocenter)

E A → ⋅ B C → + E B → ⋅ C A → + E C → ⋅ A B → = 0 (\displaystyle (\overrightarrow (EA))\cdot (\overrightarrow (BC))+(\overrightarrow (EB))\cdot (\ overrightarrow (CA))+(\overrightarrow (EC))\cdot (\overrightarrow (AB))=0)

(To prove the identity, you should use the formulas

A B → = E B → − E A → , B C → = E C → − E B → , C A → = E A → − E C → (\displaystyle (\overrightarrow (AB))=(\overrightarrow (EB))-(\overrightarrow (EA )),\,(\overrightarrow (BC))=(\overrightarrow (EC))-(\overrightarrow (EB)),\,(\overrightarrow (CA))=(\overrightarrow (EA))-(\overrightarrow (EC)))

Point E should be taken as the intersection of two altitudes of the triangle.)

• Orthocenter isogonally conjugate to the center circumscribed circle .
• Orthocenter lies on the same line as the centroid, the center circumcircle and the center of a circle of nine points (see Euler’s straight line).
• Orthocenter of an acute triangle is the center of the circle inscribed in its orthotriangle.
• The center of a triangle described by the orthocenter with vertices at the midpoints of the sides of the given triangle. The last triangle is called the complementary triangle to the first triangle.
• The last property can be formulated as follows: The center of the circle circumscribed about the triangle serves orthocenter additional triangle.
• Points, symmetrical orthocenter of a triangle with respect to its sides lie on the circumcircle.
• Points, symmetrical orthocenter triangles relative to the midpoints of the sides also lie on the circumscribed circle and coincide with points diametrically opposite to the corresponding vertices.
• If O is the center of the circumcircle ΔABC, then O H → = O A → + O B → + O C → (\displaystyle (\overrightarrow (OH))=(\overrightarrow (OA))+(\overrightarrow (OB))+(\overrightarrow (OC))) ,
• The distance from the vertex of the triangle to the orthocenter is twice as great as the distance from the center of the circumcircle to the opposite side.
• Any segment drawn from orthocenter Before intersecting with the circumcircle, it is always divided in half by the Euler circle. Orthocenter is the homothety center of these two circles.
• Hamilton's theorem. Three straight line segments connecting the orthocenter with the vertices of an acute triangle split it into three triangles having the same Euler circle (circle of nine points) as the original acute triangle.
• Corollaries of Hamilton's theorem:
  • Three straight line segments connecting the orthocenter with the vertices of an acute triangle divide it into three Hamilton triangle having equal radii of circumscribed circles.
  • The radii of circumscribed circles of three Hamilton triangles equal to the radius of the circle circumscribed about the original acute triangle.
• In an acute triangle, the orthocenter lies inside the triangle; in an obtuse angle - outside the triangle; in a rectangular one - at the vertex of a right angle.

Properties of altitudes of an isosceles triangle

• If two altitudes in a triangle are equal, then the triangle is isosceles (the Steiner-Lemus theorem), and the third altitude is both the median and the bisector of the angle from which it emerges.
• The converse is also true: in an isosceles triangle, two altitudes are equal, and the third altitude is both the median and the bisector.
• An equilateral triangle has all three heights equal.

Properties of the bases of altitudes of a triangle

• Grounds heights form a so-called orthotriangle, which has its own properties.
• The circle circumscribed about an orthotriangle is the Euler circle. This circle also contains three midpoints of the sides of the triangle and three midpoints of three segments connecting the orthocenter with the vertices of the triangle.
• Another formulation of the last property:
  • Euler's theorem for a circle of nine points. Grounds three heights arbitrary triangle, the midpoints of its three sides ( the foundations of its internal medians) and the midpoints of three segments connecting its vertices with the orthocenter, all lie on the same circle (on nine point circle).
• Theorem. In any triangle, the segment connecting grounds two heights triangle, cuts off a triangle similar to the given one.
• Theorem. In a triangle, the segment connecting grounds two heights triangles lying on two sides antiparallel to a third party with whom he has no common ground. A circle can always be drawn through its two ends, as well as through the two vertices of the third mentioned side.

Other properties of triangle altitudes

• If a triangle versatile (scalene), then it internal the bisector drawn from any vertex lies between internal median and height drawn from the same vertex.
• The height of a triangle is isogonally conjugate to the diameter (radius) circumscribed circle, drawn from the same vertex.
• In an acute triangle there are two heights cut off similar triangles from it.
• In a right triangle height, drawn from the vertex of a right angle, splits it into two triangles similar to the original one.

Properties of the minimum altitude of a triangle

The minimum altitude of a triangle has many extreme properties. For example:

• The minimum orthogonal projection of a triangle onto lines lying in the plane of the triangle has a length equal to the smallest of its altitudes.
• The minimum straight cut in a plane through which a rigid triangular plate can be pulled must have a length equal to the smallest of the heights of this plate.
• With the continuous movement of two points along the perimeter of the triangle towards each other, the maximum distance between them during the movement from the first meeting to the second cannot be less than the length of the smallest height of the triangle.
• The minimum height in a triangle always lies within that triangle.

Basic relationships

• h a = b ⋅ sin ⁡ γ = c ⋅ sin ⁡ β , (\displaystyle h_(a)=b(\cdot )\sin \gamma =c(\cdot )\sin \beta ,)
• h a = 2 ⋅ S a , (\displaystyle h_(a)=(\frac (2(\cdot )S)(a)),) Where S (\displaystyle S)- area of ​​a triangle, a (\displaystyle a)- the length of the side of the triangle by which the height is lowered.
• h a = b ⋅ c 2 ⋅ R , (\displaystyle h_(a)=(\frac (b(\cdot )c)(2(\cdot )R)),) Where b ⋅ c (\displaystyle b(\cdot )c)- product of the sides, R − (\displaystyle R-) circumscribed circle radius
• h a: h b: h c = 1 a: 1 b: 1 c = (b ⋅ c) : (a ⋅ c) : (a ⋅ b) . (\displaystyle h_(a):h_(b):h_(c)=(\frac (1)(a)):(\frac (1)(b)):(\frac (1)(c)) =(b(\cdot )c):(a(\cdot )c):(a(\cdot )b).)
• 1 h a + 1 h b + 1 h c = 1 r (\displaystyle (\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_ (c)))=(\frac (1)(r))), Where r (\displaystyle r)- radius of the inscribed circle.
• S = 1 (1 h a + 1 h b + 1 h c) ⋅ (1 h a + 1 h b − 1 h c) ⋅ (1 h a + 1 h c − 1 h b) ⋅ (1 h b + 1 h c − 1 h a) (\displaystyle S =(\frac (1)(\sqrt (((\frac (1)(h_(a)))+(\frac (1)(h_(b)))+(\frac (1)(h_(c ))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(b)))-(\frac (1)(h_(c))) )(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(c)))-(\frac (1)(h_(b))))(\ cdot )((\frac (1)(h_(b)))+(\frac (1)(h_(c)))-(\frac (1)(h_(a)))))))), Where S (\displaystyle S)- area of ​​a triangle.
• a = 2 h a ⋅ (1 h a + 1 h b + 1 h c) ⋅ (1 h a + 1 h b − 1 h c) ⋅ (1 h a + 1 h c − 1 h b) ⋅ (1 h b + 1 h c − 1 h a) (\ displaystyle a=(\frac (2)(h_(a)(\cdot )(\sqrt (((\frac (1)(h_(a)))+(\frac (1)(h_(b))) +(\frac (1)(h_(c))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(b)))-(\ frac (1)(h_(c))))(\cdot )((\frac (1)(h_(a)))+(\frac (1)(h_(c)))-(\frac (1 )(h_(b))))(\cdot )((\frac (1)(h_(b)))+(\frac (1)(h_(c)))-(\frac (1)(h_ (a))))))))), a (\displaystyle a)- the side of the triangle to which the height descends h a (\displaystyle h_(a)).
• Height of an isosceles triangle lowered to the base: h c = 1 2 ⋅ 4 a 2 − c 2 , (\displaystyle h_(c)=(\frac (1)(2))(\cdot )(\sqrt (4a^(2)-c^(2)) ),)

Where c (\displaystyle c)- base, a (\displaystyle a)- side.

Right Triangle Altitude Theorem

If the altitude in a right triangle ABC is of length h (\displaystyle h) drawn from the vertex of a right angle, divides the hypotenuse with length c (\displaystyle c) into segments m (\displaystyle m) And n (\displaystyle n), corresponding to the legs b (\displaystyle b) And a (\displaystyle a), then the following equalities are true.

Properties

• The altitudes of a triangle intersect at one point, called the orthocenter. - This statement is easy to prove using a vector identity that is valid for any points A, B, C, E, not necessarily even those lying in the same plane:

(To prove the identity, you should use the formulas

Point E should be taken as the intersection of two altitudes of the triangle.)

• In a right triangle, the altitude drawn from the vertex of the right angle splits it into two triangles similar to the original one.
• In an acute triangle, its two altitudes cut off similar triangles from it.
• The bases of the heights form a so-called orthotriangle, which has its own properties.

The minimum altitude of a triangle has many extreme properties. For example:

• The minimum orthogonal projection of a triangle onto lines lying in the plane of the triangle has a length equal to the smallest of its altitudes.

• The minimum straight cut in a plane through which a rigid triangular plate can be pulled must have a length equal to the smallest of the heights of this plate.

• With the continuous movement of two points along the perimeter of the triangle towards each other, the maximum distance between them during the movement from the first meeting to the second cannot be less than the length of the smallest height of the triangle.

The minimum height in a triangle always lies within that triangle.

Basic relationships

where is the area of ​​the triangle, is the length of the side of the triangle by which the height is lowered.

where is the base.

Right Triangle Altitude Theorem

If a height of length h drawn from the vertex of a right angle divides the hypotenuse of length c into segments m and n corresponding to b and a, then the following equalities are true:

Mnemonic poem

The height is like a cat, Which, arching its back, And at right angles Connects the top And the side with its tail.

see also

Links

Wikimedia Foundation. 2010.

See what “Height of a triangle” is in other dictionaries:

HEIGHT, heights, plural. heights, heights, women 1. units only Extension from bottom to top, height. Height of the house. Tower of great height. || (pl. only special scientific). Distance from the earth's surface, measured along a vertical line from bottom to top. The airplane was flying... Ushakov's Explanatory Dictionary

This term has other meanings, see Height (meanings). Height in elementary geometry is a perpendicular segment lowered from the top of a geometric figure (for example, a triangle, pyramid, cone) to its base or to ... ... Wikipedia

height- ы/; pl. height/you; and. see also high-rise, high-rise 1) Size, length of something. from the bottom to the top, from bottom to top. Height/ of a house, tree, mountain. Height/waves. The dam is one hundred five feet high... Dictionary of many expressions

Y; pl. heights; and. 1. Size, length of something. from the bottom to the top, from bottom to top. V. houses, trees, mountains. V. waves. The dam is one hundred and fifty meters high. Measure, determine the height of something. 2. Distance from which l. surface to... ... encyclopedic Dictionary

height of original thread triangle- (H) The distance between the apex and base of the original thread triangle in a direction perpendicular to the axis of the thread. [GOST 11708 82 (ST SEV 2631 80)] Topics of the interchangeability standard General terms basic elements and thread parameters EN ... ... Technical Translator's Guide

Height is the dimension or distance in the vertical direction. Other meanings: In astronomy: The height of the luminary is the angle between the plane of the mathematical horizon and the direction towards the luminary. In military affairs: Height is the elevation of the relief. In... ... Wikipedia

HEIGHT, in geometry, a perpendicular segment descended from the top of a geometric figure (e.g., triangle, pyramid, cone) to its base (or continuation of the base), as well as the length of this segment. The height of a prism, cylinder, spherical layer, and... ... encyclopedic Dictionary

In geometry, a perpendicular segment drawn from the top of a geometric figure (e.g., triangle, pyramid, cone) to its base (or continuation of the base), as well as the length of this segment. The height of the prism, cylinder, spherical layer, as well as... ... Big Encyclopedic Dictionary

HEIGHT, s, plural. from, from, from, wives. 1. Size, length of something. from the bottom point to the top. B. brickwork. V. surf. V. cyclone. 2. Space, distance from the ground up. Look up. The plane is gaining altitude. Fly to... ... Ozhegov's Explanatory Dictionary

Height in geometry, a perpendicular segment descended from the top of a geometric figure (for example, a triangle, pyramid, cone) to its base or continuation of the base, as well as the length of this segment. B. prism, cylinder, spherical layer,... ... Great Soviet Encyclopedia

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A triangle is a polygon with three sides, or a closed broken line with three links, or a figure formed by three segments connecting three points that do not lie on the same straight line (see Fig. 1).

Basic elements of triangle abc

Peaks – points A, B, and C;

Parties – segments a = BC, b = AC and c = AB connecting the vertices;

Angles – α, β, γ formed by three pairs of sides. Angles are often designated in the same way as vertices, with the letters A, B, and C.

The angle formed by the sides of a triangle and lying in its interior area is called an interior angle, and the one adjacent to it is the adjacent angle of the triangle (2, p. 534).

Heights, medians, bisectors and midlines of a triangle

In addition to the main elements in a triangle, other segments with interesting properties are also considered: heights, medians, bisectors and midlines.

Height

Triangle heights- these are perpendiculars dropped from the vertices of the triangle to opposite sides.

To plot the height, you must perform the following steps:

1) draw a straight line containing one of the sides of the triangle (if the height is drawn from the vertex of an acute angle in an obtuse triangle);

2) from the vertex lying opposite the drawn line, draw a segment from the point to this line, making an angle of 90 degrees with it.

The point where the altitude intersects the side of the triangle is called height base (see Fig. 2).

Properties of triangle altitudes

In a right triangle, the altitude drawn from the vertex of the right angle splits it into two triangles similar to the original triangle.

In an acute triangle, its two altitudes cut off similar triangles from it.

If the triangle is acute, then all the bases of the altitudes belong to the sides of the triangle, and in an obtuse triangle, two altitudes fall on the continuation of the sides.

Three altitudes in an acute triangle intersect at one point and this point is called orthocenter triangle.

Median

Medians(from Latin mediana – “middle”) - these are segments connecting the vertices of the triangle with the midpoints of the opposite sides (see Fig. 3).

To construct the median, you must perform the following steps:

1) find the middle of the side;

2) connect the point that is the middle of the side of the triangle with the opposite vertex with a segment.

Properties of triangle medians

The median divides a triangle into two triangles of equal area.

The medians of a triangle intersect at one point, which divides each of them in a ratio of 2:1, counting from the vertex. This point is called center of gravity triangle.

The entire triangle is divided by its medians into six equal triangles.

Bisector

Bisectors(from Latin bis - twice and seko - cut) are the straight line segments enclosed inside a triangle that bisect its angles (see Fig. 4).

To construct a bisector, you must perform the following steps:

1) construct a ray coming out from the vertex of the angle and dividing it into two equal parts (the bisector of the angle);

2) find the point of intersection of the bisector of the angle of the triangle with the opposite side;

3) select a segment connecting the vertex of the triangle with the intersection point on the opposite side.

Properties of triangle bisectors

The bisector of an angle of a triangle divides the opposite side in a ratio equal to the ratio of the two adjacent sides.

The bisectors of the interior angles of a triangle intersect at one point. This point is called the center of the inscribed circle.

The bisectors of the internal and external angles are perpendicular.

If the bisector of an exterior angle of a triangle intersects the extension of the opposite side, then ADBD=ACBC.

The bisectors of one internal and two external angles of a triangle intersect at one point. This point is the center of one of the three excircles of this triangle.

The bases of the bisectors of two internal and one external angles of a triangle lie on the same straight line if the bisector of the external angle is not parallel to the opposite side of the triangle.

If the bisectors of the external angles of a triangle are not parallel to opposite sides, then their bases lie on the same straight line.