Corner side 2 3 base height. Triangle Height

When solving geometric problems, it is useful to follow this algorithm. While reading the task statement, it is necessary

Make a drawing. The drawing should correspond as much as possible to the condition of the problem, so its main task is to help find the solution
Apply all the data from the task condition to the drawing
Write out all the geometric concepts that occur in the problem
Recall all the theorems that relate to this concept
Put on the drawing all the relationships between the elements of a geometric figure that follow from these theorems

For example, if the task contains the words bisector of the angle of a triangle, you need to remember the definition and properties of the bisector and designate equal or proportional segments and angles in the drawing.

In this article, you will find the basic properties of a triangle that you need to know to successfully solve problems.

TRIANGLE.

Area of a triangle.

1. ,

here - an arbitrary side of the triangle, - the height lowered to this side.

2. ,

here and are arbitrary sides of the triangle, is the angle between these sides:

3. Heron formula:

Here - the lengths of the sides of the triangle, - the semiperimeter of the triangle,

4. ,

here - the semiperimeter of the triangle, - the radius of the inscribed circle.

Let be the lengths of the tangent segments.

Then Heron's formula can be written in the following form:

6. ,

here - the lengths of the sides of the triangle, - the radius of the circumscribed circle.

If a point is taken on a side of a triangle that divides this side in the ratio m:n, then the segment connecting this point with the vertex of the opposite angle divides the triangle into two triangles, the areas of which are related as m:n:

The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

Triangle median

This is a line segment that connects the vertex of the triangle with the midpoint of the opposite side.

Triangle medians intersect at one point and share the intersection point in a ratio of 2:1, counting from the top.

The intersection point of the medians of a regular triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger one is equal to the radius of the circumscribed circle.

The radius of the circumscribed circle is twice the radius of the inscribed circle: R=2r

Median length arbitrary triangle

here - the median drawn to the side - the lengths of the sides of the triangle.

Bisector of a triangle

This is a segment of the bisector of any angle of a triangle, connecting the vertex of this angle with the opposite side.

Bisector of a triangle divides the side into segments proportional to the adjacent sides:

Triangle bisectors intersect at one point, which is the center of the inscribed circle.

All points on the bisector of an angle are equidistant from the sides of the angle.

Triangle Height

This is a segment of the perpendicular, lowered from the vertex of the triangle to opposite side, or its continuation. In an obtuse triangle, the altitude drawn from the vertex of an acute angle lies outside the triangle.

The heights of a triangle intersect at one point, which is called the triangle's orthocenter.

To find the height of a triangle drawn to the side, you need to find its area in any way possible, and then use the formula:

Center of a circle circumscribed about a triangle, lies at the point of intersection of the perpendicular bisectors drawn to the sides of the triangle.

The radius of the circumscribed circle of a triangle can be found using the following formulas:

Here are the lengths of the sides of the triangle, area of a triangle.

where is the length of the side of the triangle, is the opposite angle. (This formula follows from the sine theorem).

triangle inequality

Each side of the triangle is less than the sum and greater than the difference of the other two.

The sum of the lengths of any two sides is always greater than the length of the third side:

Opposite the larger side lies a larger angle; opposite the larger angle lies the larger side:

If , then vice versa.

Sine theorem:

The sides of a triangle are proportional to the sines of the opposite angles:

Cosine theorem:

the square of a side of a triangle is equal to the sum of the squares of the other two sides without doubling the product of these sides by the cosine of the angle between them:

Right triangle

- It is a triangle with one of the angles equal to 90°.

Sum sharp corners right triangle equals 90°.

The hypotenuse is the side that lies opposite the 90° angle. The hypotenuse is the longest side.

Pythagorean theorem:

the square of the hypotenuse is equal to the sum of the squares of the legs:

The radius of a circle inscribed in a right triangle is

here - the radius of the inscribed circle, - the legs, - the hypotenuse:

Center of a circle circumscribed about a right triangle lies in the middle of the hypotenuse:

Median of a right triangle drawn to the hypotenuse equal to half of the hypotenuse.

Definition of sine, cosine, tangent and cotangent of a right triangle see

The ratio of elements in a right triangle:

The square of the altitude of a right triangle drawn from a vertex right angle, is equal to the product of the projections of the legs on the hypotenuse:

The square of the leg is equal to the product of the hypotenuse and the projection of the leg to the hypotenuse:

Leg lying against the corner equal to half the hypotenuse:

Isosceles triangle.

Bisector isosceles triangle drawn to the base is the median and the height.

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

Top angle.

I - sides

And - angles at the base.

Height, bisector and median.

Attention! The height, bisector and median drawn to the lateral side do not match.

right triangle

(or equilateral triangle ) is a triangle, all sides and angles of which are equal to each other.

Area of an equilateral triangle is equal to

where is the length of the side of the triangle.

Center of a circle inscribed in an equilateral triangle, coincides with the center of the circle circumscribed about an equilateral triangle and lies at the point of intersection of the medians.

Intersection point of medians of an equilateral triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger one is equal to the radius of the circumscribed circle.

If one of the angles of an isosceles triangle is 60°, then the triangle is regular.

Middle line of the triangle

This is a segment that connects the midpoints of two sides.

In the figure, DE is the midline of triangle ABC.

The midline of the triangle is parallel to the third side and equal to half of it: DE||AC, AC=2DE

External corner of a triangle

This is the angle adjacent to any angle of the triangle.

An exterior angle of a triangle is equal to the sum of two angles not adjacent to it.

Trigonometric functions of an external angle:

Signs of equality of triangles:

1 . If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.

2 . If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.

3 If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

Important: since in a right triangle two angles are obviously equal, then for equality of two right triangles only two elements are required to be equal: two sides, or a side and an acute angle.

Signs of similarity of triangles:

1 . If two sides of one triangle are proportional to two sides of another triangle, and the angles included between these sides are equal, then these triangles are similar.

2 . If three sides of one triangle are proportional to three sides of another triangle, then these triangles are similar.

3 . If two angles of one triangle are equal to two angles of another triangle, then these triangles are similar.

Important: In similar triangles, similar sides lie opposite equal angles.

Theorem of Menelaus

Let the line intersect the triangle , where is the point of its intersection with the side , is the point of its intersection with the side , and is the point of its intersection with the extension of the side . Then

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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, one 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)))

The point E should be taken as the intersection of the two heights of the triangle.)

Orthocenter isogonally conjugate to the center circumscribed circle .
Orthocenter lies on the same line as the centroid, the center circumscribed circle and the center of the circle of nine points (see the Euler line).
Orthocenter an acute triangle is the center of a 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 an additional triangle with respect to the first triangle.
The last property can be formulated as follows: The center of a circle circumscribed about a triangle serves orthocenter additional triangle.
Points, symmetrical orthocenter triangle with respect to its sides lie on the circumscribed circle.
Points, symmetrical orthocenter triangles with respect to the midpoints of the sides also lie on the circumscribed circle and coincide with points diametrically opposite to the corresponding vertices.
If ABOUT is the center of the circumscribed circle 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 the distance from the center of the circumscribed circle to the opposite side.
Any segment drawn from orthocenter always bisects the Euler circle until it intersects with the circumscribed circle. Orthocenter is the center of the homothety of these two circles.
Hamilton's theorem. Three line segments connecting the orthocenter with the vertices of an acute-angled triangle divide it into three triangles having the same Euler circle (circle of nine points) as the original acute-angled triangle.
Consequences of Hamilton's theorem:
- Three line segments connecting the orthocenter with the vertices of an acute-angled triangle divide it into three Hamilton triangle having equal radii of circumscribed circles.
- The radii of the circumscribed circles of the three Hamilton triangles are equal to the radius of the circle circumscribed about the original acute-angled triangle.
In an acute triangle, the orthocenter lies inside the triangle; in obtuse - outside the triangle; in a rectangular one - at the vertex of a right angle.

Elevation Properties of an Isosceles Triangle

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

Properties of the bases of the heights of a triangle

Foundations heights form the so-called orthotriangle, which has its own properties.
The circle described near the orthotriangle is the Euler circle. Three midpoints of the sides of the triangle and three midpoints of the three segments connecting the orthocenter with the vertices of the triangle also lie on this circle.
Another formulation of the last property:
- Euler's theorem for a circle of nine points. Foundations three heights arbitrary triangle, the midpoints of its three sides ( foundations of its internal medians) and the midpoints of the three segments connecting its vertices with the orthocenter, all lie on the same circle (on nine point circle).
Theorem. In any triangle, the line segment connecting grounds two heights triangle cuts off a triangle similar to the given one.
Theorem. In a triangle, the line segment connecting grounds two heights triangles on two sides antiparallel a third party with whom he has no common points. Through its two ends, as well as through two vertices of the third mentioned side, it is always possible to draw a circle.

Other properties of triangle heights

Properties of the minimum height of a triangle

The minimum height 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 heights.
The minimum straight cut in the plane through which an inflexible triangular plate can be pulled must have a length equal to the smallest of the heights of this plate.
With 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 of the heights of the triangle.
The minimum height in a triangle is always within that triangle.

Basic ratios

h a = b sin ⁡ γ = c sin ⁡ β , (\displaystyle h_(a)=b\sin \gamma =c\sin \beta ,)
h a = 2 S a , (\displaystyle h_(a)=(\frac (2S)(a)),) Where S (\displaystyle S)- area of a triangle, a (\displaystyle a)- the length of the side of the triangle on which the height is lowered.
h a 2 = 1 2 (b 2 + c 2 − 1 2 (a 2 + (b 2 − c 2) 2 a 2)) (\displaystyle h_(a)^(2)=(\frac (1)(2 ))(b^(2)+c^(2)-(\frac (1)(2))(a^(2)+(\frac ((b^(2)-c^(2))^ (2))(a^(2))))))
h a = b c 2 R , (\displaystyle h_(a)=(\frac (bc)(2R)),) Where bc (\displaystyle bc)- the product of the sides, R − (\displaystyle R-) radius of the circumscribed circle
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))=bc:ac:ab)
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) is the 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 falls h a (\displaystyle h_(a)).
The height of an isosceles triangle lowered to the base: h c = 1 2 4 a 2 − c 2 , (\displaystyle h_(c)=(\frac (1)(2))(\sqrt (4a^(2)-c^(2))),)

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

Theorem on the height of a right triangle

If the height in a right triangle A B C (\displaystyle ABC) long h (\displaystyle h), drawn from the vertex of a right angle, divides the hypotenuse with a 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.

To solve many geometric problems, you need to find the height of a given figure. These tasks have applied value. When carrying out construction work, determining the height helps to calculate required amount materials, as well as determine how accurately slopes and openings are made. Often, to build patterns, you need to have an idea about the properties

Many people, despite good grades at school, when building ordinary geometric shapes the question arises of how to find the height of a triangle or parallelogram. And it is the most difficult. This is because a triangle can be acute, obtuse, isosceles, or right. Each of them has its own rules for construction and calculation.

How to find the height of a triangle in which all angles are acute, graphically

If all the angles of the triangle are acute (each angle in the triangle is less than 90 degrees), then to find the height, do the following.

According to the given parameters, we construct a triangle.
Let us introduce notation. A, B and C will be the vertices of the figure. The angles corresponding to each vertex are α, β, γ. The sides opposite these corners are a, b, c.
The height is the perpendicular from the vertex of the angle to the opposite side of the triangle. To find the heights of a triangle, we construct perpendiculars: from the vertex of angle α to side a, from the vertex of angle β to side b, and so on.
The intersection point of the height and side a will be denoted by H1, and the height itself will be h1. The intersection point of the height and side b will be H2, the height, respectively, h2. For side c, the height will be h3 and the intersection point H3.

Height in a triangle with an obtuse angle

Now consider how to find the height of a triangle if one (greater than 90 degrees). In this case, the height drawn from an obtuse angle will be inside the triangle. The remaining two heights will be outside the triangle.

Let the angles α and β in our triangle be acute, and the angle γ be obtuse. Then, to construct the heights coming out of the angles α and β, it is necessary to continue the sides of the triangle opposite to them in order to draw perpendiculars.

How to find the height of an isosceles triangle

This figure has two equal sides and the base, while the angles at the base are also equal to each other. This equality of sides and angles facilitates the construction of heights and their calculation.

First, let's draw the triangle itself. Let the sides b and c, as well as the angles β, γ be respectively equal.

Now let's draw a height from the vertex of the angle α, denote it h1. For this height will be both the bisector and the median.

Only one construction can be made for the foundation. For example, draw a median - a segment connecting the vertex of an isosceles triangle and the opposite side, the base, to find the height and bisector. And to calculate the length of the height for the other two sides, you can build only one height. Thus, in order to graphically determine how to calculate the height of an isosceles triangle, it is enough to find two heights out of three.

How to find the height of a right triangle

It is much easier to determine the heights of a right triangle than others. This is because the legs themselves make up a right angle, which means they are heights.

To build the third height, as usual, a perpendicular is drawn connecting the vertex of the right angle and the opposite side. As a result, in order to make a triangle in this case, only one construction is required.