home
Secrets of the world
Acute-angled isosceles. What is a triangle

Acute-angled isosceles. What is a triangle

29.09.2019

triangles

triangle A figure is called a figure that consists of three points that do not lie on one straight line, and three segments connecting these points in pairs. The points are called peaks triangle, and the segments - its parties.

Types of triangles

The triangle is called isosceles if its two sides are equal. These equal sides are called sides, and the third party is called basis triangle.

A triangle in which all sides are equal is called equilateral or correct.

The triangle is called rectangular, if it has a right angle, then there is a 90° angle. The side of a right triangle opposite the right angle is called hypotenuse the other two sides are called legs.

The triangle is called acute-angled if all three of its angles are acute, that is, less than 90 °.

The triangle is called obtuse, if one of its angles is obtuse, i.e. greater than 90°.

The main lines of the triangle

Median

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

Triangle median properties

The median divides the triangle into two triangles of the same area.

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

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

Bisector

Angle bisector is a ray that comes from its vertex, passes between its sides and bisects the given angle. Bisector of a triangle A segment of the bisector of an angle of a triangle connecting a vertex to a point on the opposite side of the triangle is called.

Triangle bisector properties

Height

Height triangle is called a perpendicular drawn from the vertex of the triangle to the line containing the opposite side of this triangle.

Triangle height properties

IN right triangle the height drawn from the vertex of a right angle divides it into two triangles, similar original.

IN acute triangle its two heights cut off from it similar triangles.

Median perpendicular

A line passing through the midpoint of a segment perpendicular to it is called perpendicular bisector to the segment .

Properties of the perpendicular bisectors of a triangle

Each point of the perpendicular bisector to a segment is equidistant from the ends of this segment. The converse statement is also true: each point equidistant from the ends of the segment lies on the perpendicular bisector to it.

The point of intersection of the midperpendiculars drawn to the sides of the triangle is the center circle circumscribed about this triangle.

middle line

The middle line of the triangle A line segment joining the midpoints of two of its sides is called.

Property of the midline of a triangle

The midline of a triangle is parallel to one of its sides and equal to half of that side.

Formulas and ratios

Signs of equality of triangles

Two triangles are congruent if they are respectively congruent:

two sides and the angle between them;

two corners and a side adjacent to them;

three sides.

Signs of equality of right triangles

Two right triangle are equal if they are respectively equal:

hypotenuse and acute angle

leg and the opposite corner;

leg and adjacent angle;

two leg;

hypotenuse And leg.

similarity of triangles

Two triangles are similar if one of the following conditions is met, called signs of similarity:

two angles of one triangle are equal to two angles of another triangle;

two sides of one triangle are proportional to two sides of another triangle, and the angles formed by these sides are equal;

the three sides of one triangle are respectively proportional to the three sides of the other triangle.

In similar triangles, the corresponding lines ( heights, medians, bisectors etc.) are proportional.

Sine theorem

The sides of a triangle are proportional to the sines of the opposite angles, and the coefficient of proportionality is equal to diameter circle circumscribed about a triangle:

Cosine theorem

The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them:

a 2 = b 2 + c 2 - 2bc cos

Triangle area formulas

Arbitrary triangle

a, b, c - sides; - angle between sides a And b; - semi-perimeter; R- radius of the circumscribed circle; r- radius of the inscribed circle; S- square; h a - height to side a.

Even preschool children know what a triangle looks like. But with what they are, the guys are already starting to understand at school. One type is an obtuse triangle. To understand what it is, the easiest way is to see a picture with its image. And in theory, this is what they call the "simplest polygon" with three sides and vertices, one of which is

Understanding concepts

In geometry, there are such types of figures with three sides: acute-angled, right-angled and obtuse-angled triangles. Moreover, the properties of these simplest polygons are the same for all. So, for all the listed species, such an inequality will be observed. The sum of the lengths of any two sides is necessarily greater than the length of the third side.

But in order to be sure that we are talking about a complete figure, and not about a set of individual vertices, it is necessary to check that the main condition is met: the sum of the angles of an obtuse triangle is 180 o. The same is true for other types of figures with three sides. True, in an obtuse triangle one of the angles will be even more than 90 o, and the remaining two will necessarily be sharp. In this case, it is the largest angle that will be opposite the longest side. True, these are far from all the properties of an obtuse triangle. But even knowing only these features, students can solve many problems in geometry.

For every polygon with three vertices, it is also true that by continuing any of the sides, we get an angle whose size will be equal to the sum of two non-adjacent internal vertices. The perimeter of an obtuse triangle is calculated in the same way as for other shapes. It is equal to the sum of the lengths of all its sides. To determine the mathematicians, various formulas were derived, depending on what data was initially present.

Correct style

One of the most important conditions for solving problems in geometry is the correct drawing. Mathematics teachers often say that it will help not only visualize what is given and what is required of you, but also get 80% closer to the correct answer. That is why it is important to know how to construct an obtuse triangle. If you just want a hypothetical figure, then you can draw any polygon with three sides so that one of the angles is greater than 90 degrees.

If certain values of the lengths of the sides or degrees of angles are given, then it is necessary to draw an obtuse-angled triangle in accordance with them. At the same time, it is necessary to try to depict the angles as accurately as possible, calculating them with the help of a protractor, and display the sides in proportion to the given conditions in the task.

Main lines

Often, it is not enough for schoolchildren to know only how certain figures should look. They cannot limit themselves to information about which triangle is obtuse and which is right-angled. The course of mathematics provides that their knowledge of the main features of the figures should be more complete.

So, each student should understand the definition of the bisector, median, perpendicular bisector and height. In addition, he must know their basic properties.

So, the bisectors divide the angle in half, and the opposite side into segments that are proportional to the adjacent sides.

The median divides any triangle into two equal areas. At the point at which they intersect, each of them is divided into 2 segments in a ratio of 2: 1, when viewed from the top from which it originated. In this case, the largest median is always drawn to its smallest side.

No less attention is paid to height. This is perpendicular to the opposite side from the corner. The height of an obtuse triangle has its own characteristics. If it is drawn from a sharp vertex, then it falls not on the side of this simplest polygon, but on its extension.

The perpendicular bisector is the line segment that comes out of the center of the face of the triangle. At the same time, it is located at a right angle to it.

Working with circles

At the beginning of the study of geometry, it is enough for children to understand how to draw an obtuse-angled triangle, learn to distinguish it from other types and remember its basic properties. But for high school students this knowledge is not enough. For example, at the exam, there are often questions about circumscribed and inscribed circles. The first of them touches all three vertices of the triangle, and the second has one common point with all sides.

Constructing an inscribed or circumscribed obtuse-angled triangle is already much more difficult, because for this you first need to find out where the center of the circle and its radius should be. By the way, in this case, not only a pencil with a ruler, but also a compass will become a necessary tool.

The same difficulties arise when constructing inscribed polygons with three sides. Mathematicians have developed various formulas that allow you to determine their location as accurately as possible.

Inscribed Triangles

As mentioned earlier, if the circle passes through all three vertices, then this is called the circumscribed circle. Its main property is that it is the only one. To find out how the circumscribed circle of an obtuse triangle should be located, it must be remembered that its center is at the intersection of the three median perpendiculars that go to the sides of the figure. If in an acute-angled polygon with three vertices this point will be inside it, then in an obtuse-angled one - outside it.

Knowing, for example, that one of the sides of an obtuse triangle is equal to its radius, one can find the angle that lies opposite the known face. Its sine will be equal to the result of dividing the length of the known side by 2R (where R is the radius of the circle). That is, the sin of the angle will be equal to ½. So the angle will be 150 o.

If you need to find the radius of the circumscribed circle of an obtuse-angled triangle, then you will need information about the length of its sides (c, v, b) and its area S. After all, the radius is calculated as follows: (c x v x b): 4 x S. By the way, it doesn’t matter what kind of figure do you have: a versatile obtuse triangle, isosceles, right or acute. In any situation, thanks to the above formula, you can find out the area of a given polygon with three sides.

Circumscribed Triangles

It is also quite common to work with inscribed circles. According to one of the formulas, the radius of such a figure, multiplied by ½ of the perimeter, will equal the area of the triangle. True, to find it out, you need to know the sides of an obtuse triangle. Indeed, in order to determine ½ of the perimeter, it is necessary to add their lengths and divide by 2.

To understand where the center of a circle inscribed in an obtuse triangle should be, it is necessary to draw three bisectors. These are the lines that bisect the corners. It is at their intersection that the center of the circle will be located. In this case, it will be equidistant from each side.

The radius of such a circle inscribed in an obtuse triangle is equal to the quotient (p-c) x (p-v) x (p-b) : p. Moreover, p is the half-perimeter of the triangle, c, v, b are its sides.

The division of triangles into acute, right and obtuse triangles. Classification by aspect ratio divides triangles into scalene, equilateral and isosceles. Moreover, each triangle simultaneously belongs to two. For example, it can be rectangular and versatile at the same time.

When determining the type by the type of corners, be very careful. An obtuse-angled triangle will be called such a triangle, in which one of the angles is, that is, it is more than 90 degrees. A right triangle can be calculated by having one right (equal to 90 degrees) angle. However, to classify a triangle as an acute triangle, you will need to make sure that all three of its angles are acute.

Defining the view triangle by aspect ratio, first you have to find out the lengths of all three sides. However, if by condition the lengths of the sides are not given to you, the angles can help you. A triangle will be versatile, all three sides of which have different lengths. If the lengths of the sides are unknown, then a triangle can be classified as scalene if all three of its angles are different. A scalene triangle can be obtuse, right-angled or acute-angled.

A triangle is isosceles if two of its three sides are equal. If the lengths of the sides are not given to you, be guided by two equal angles. An isosceles triangle, like a scalene one, can be obtuse, right-angled and acute-angled.

An equilateral triangle can only be such that all three sides of which have the same length. All its angles are also equal to each other, and each of them is equal to 60 degrees. From this it is clear that equilateral triangles are always acute-angled.

Advice 2: How to identify an obtuse and acute triangle

The simplest of the polygons is the triangle. It is formed with the help of three points lying in the same plane, but not lying on the same straight line, connected in pairs by segments. However, triangles come in different types, which means they have different properties.

Instruction

It is customary to distinguish three types: obtuse, acute and rectangular. It's like the corners. An obtuse triangle is a triangle in which one of the angles is obtuse. An obtuse angle is one that is greater than ninety degrees but less than one hundred and eighty. For example, in triangle ABC, angle ABC is 65°, angle BCA is 95°, and angle CAB is 20°. Angles ABC and CAB are less than 90°, but angle BCA is greater, so the triangle is obtuse.

An acute triangle is a triangle in which all angles are acute. An acute angle is one that is less than ninety and greater than zero degrees. For example, in triangle ABC, angle ABC is 60°, angle BCA is 70°, and angle CAB is 50°. All three angles are less than 90°, so it is a triangle. If you know that all sides of a triangle are equal, it means that all angles are also equal to each other, and at the same time they are equal to sixty degrees. Accordingly, all angles in such a triangle are less than ninety degrees, and therefore such a triangle is acute-angled.

If in a triangle one of the angles is equal to ninety degrees, this means that it does not belong to either the wide-angle type or the acute-angle type. This is a right triangle.

If the type of triangle is determined by the aspect ratio, they will be equilateral, scalene and isosceles. In an equilateral triangle, all sides are equal, and this, as you found out, indicates that the triangle is acute. If a triangle has only two equal sides or if the sides are not equal to each other, it can be obtuse, right-angled, or acute-angled. So, in these cases, it is necessary to calculate or measure the angles and draw conclusions, according to paragraphs 1, 2 or 3.

Signs of equality of triangles

A triangle on the Euclidean plane can be uniquely (up to congruence) defined by the following triplets of basic elements:

a, b, γ (equality on two sides and the angle lying between them);
a, β, γ (equality in side and two adjacent angles);
a, b, c (equality on three sides).

Signs of equality of right triangles:

along the leg and hypotenuse;
on two legs;
along the leg and acute angle;
hypotenuse and acute angle.

Some points in the triangle are "paired". For example, there are two points from which all sides are visible either at an angle of 60° or at an angle of 120°. They're called dots Torricelli. There are also two points whose projections on the sides lie at the vertices of a regular triangle. This - points of Apollonius. Points and such as are called Brocard points.

Direct

In any triangle, the center of gravity, the orthocenter and the center of the circumscribed circle lie on the same straight line, called Euler line.

The line passing through the center of the circumscribed circle and the Lemoine point is called Brokar's axis. Apollonius points lie on it. The Torricelli points and the Lemoine point also lie on the same straight line. The bases of the outer bisectors of the angles of a triangle lie on the same straight line, called axis of external bisectors. The points of intersection of the lines containing the sides of the orthotriangle with the lines containing the sides of the triangle also lie on the same line. This line is called orthocentric axis, it is perpendicular to the Euler line.

If we take a point on the circumscribed circle of a triangle, then its projections on the sides of the triangle will lie on one straight line, called Simson's straight line given point. Simson's lines of diametrically opposite points are perpendicular.

triangles

A triangle with vertices at the bases of cevians drawn through a given point is called cevian triangle this point.
A triangle with vertices in the projections of a given point onto the sides is called under the skin or pedal triangle this point.
A triangle with vertices at the second intersection points of lines drawn through the vertices and a given point, with a circumscribed circle, is called cevian triangle. A cevian triangle is similar to a subdermal one.

circles

Inscribed circle is a circle tangent to all three sides of the triangle. She is the only one. The center of the inscribed circle is called incenter.
Circumscribed circle- a circle passing through all three vertices of the triangle. The circumscribed circle is also unique.
Excircle- a circle tangent to one side of a triangle and the extension of the other two sides. There are three such circles in a triangle. Their radical center is the center of the inscribed circle of the median triangle, called Spieker's point.

The midpoints of the three sides of a triangle, the bases of its three altitudes, and the midpoints of the three line segments connecting its vertices to the orthocenter lie on a single circle called circle of nine points or Euler circle. The center of the nine-point circle lies on the Euler line. A circle of nine points touches an inscribed circle and three excircles. The point of contact between an inscribed circle and a circle of nine points is called Feuerbach point. If from each vertex we lay out triangles on straight lines containing sides, orthoses equal in length to opposite sides, then the resulting six points lie on one circle - Conway circles. In any triangle, three circles can be inscribed in such a way that each of them touches two sides of the triangle and two other circles. Such circles are called Malfatti circles. The centers of the circumscribed circles of the six triangles into which the triangle is divided by medians lie on one circle, which is called Lamun circle.

A triangle has three circles that touch two sides of the triangle and the circumscribed circle. Such circles are called semi-inscribed or Verrier circles. The segments connecting the points of contact of the Verrier circles with the circumscribed circle intersect at one point, called Verrier point. It serves as the center of the homothety, which takes the circumscribed circle to the incircle. The points of tangency of the Verrier circles with the sides lie on a straight line that passes through the center of the inscribed circle.

The line segments connecting the tangent points of the inscribed circle with the vertices intersect at one point, called Gergonne point, and the segments connecting the vertices with the points of contact of the excircles - in Nagel point.

Ellipses, parabolas and hyperbolas

Inscribed conic (ellipse) and its perspective

An infinite number of conics (ellipses, parabolas, or hyperbolas) can be inscribed in a triangle. If we inscribe an arbitrary conic in a triangle and connect the points of contact with opposite vertices, then the resulting lines will intersect at one point, called perspective conics. For any point of the plane that does not lie on a side or on its extension, there exists an inscribed conic with a perspective at that point.

Steiner's ellipse circumscribed and cevians passing through its foci

An ellipse can be inscribed in a triangle that touches the sides at the midpoints. Such an ellipse is called Steiner inscribed ellipse(its perspective will be the centroid of the triangle). The described ellipse, which is tangent to lines passing through vertices parallel to the sides, is called circumscribed by the Steiner ellipse. If an affine transformation ("skew") translates the triangle into a regular one, then its inscribed and circumscribed Steiner ellipse will go into an inscribed and circumscribed circle. Cevians drawn through the foci of the described Steiner ellipse (Skutin points) are equal (Skutin's theorem). Of all the circumscribed ellipses, the Steiner circumscribed ellipse has the smallest area, and of all the inscribed ellipses, the Steiner inscribed ellipse has the largest area.

Brocard's ellipse and its perspector - Lemoine point

An ellipse with foci at Brokar's points is called Brocard ellipse. Its perspective is the Lemoine point.

Properties of an inscribed parabola

Kiepert parabola

The perspectives of the inscribed parabolas lie on the circumscribed Steiner ellipse. The focus of an inscribed parabola lies on the circumscribed circle, and the directrix passes through the orthocenter. A parabola inscribed in a triangle whose directrix is the Euler line is called Kiepert's parabola. Its perspective is the fourth point of intersection of the circumscribed circle and the circumscribed Steiner ellipse, called Steiner point.

Cypert's hyperbole

If the described hyperbola passes through the intersection point of the heights, then it is equilateral (that is, its asymptotes are perpendicular). The intersection point of the asymptotes of an equilateral hyperbola lies on a circle of nine points.

Transformations

If the lines passing through the vertices and some point not lying on the sides and their extensions are reflected with respect to the corresponding bisectors, then their images will also intersect at one point, which is called isogonally conjugate the original one (if the point lay on the circumscribed circle, then the resulting lines will be parallel). Many pairs of remarkable points are isogonally conjugate: the center of the circumscribed circle and the orthocenter, the centroid and the Lemoine point, the Brocard points. The Apollonius points are isogonally conjugate to the Torricelli points, and the center of the incircle is isogonally conjugate to itself. Under the action of isogonal conjugation, straight lines go into circumscribed conics, and circumscribed conics into straight lines. Thus, the Kiepert hyperbola and the Brocard axis, the Enzhabek hyperbola and the Euler line, the Feuerbach hyperbola and the line of centers of the inscribed circle are isogonally conjugate. The circumscribed circles of subdermal triangles of isogonally conjugate points coincide. The foci of the inscribed ellipses are isogonally conjugate.

If, instead of a symmetric cevian, we take a cevian whose base is as far from the middle of the side as the base of the original one, then such cevians will also intersect at one point. The resulting transformation is called isotomic conjugation. It also maps lines to circumscribed conics. The Gergonne and Nagel points are isotomically conjugate. Under affine transformations, isotomically conjugate points pass into isotomically conjugate ones. At isotomy conjugation, the described Steiner ellipse passes into the straight line at infinity.

If, in the segments cut off by the sides of the triangle from the circumscribed circle, circles are inscribed that touch the sides at the bases of the cevians drawn through a certain point, and then the points of contact of these circles are connected to the circumscribed circle with opposite vertices, then such lines will intersect at one point. The transformation of the plane, which compares the resulting point to the starting point, is called isocircular transformation. The composition of the isogonal and isotomic conjugations is the composition of the isocircular transformation with itself. This composition is a projective transformation that leaves the sides of the triangle in place, and translates the axis of the outer bisectors into a straight line at infinity.

If we continue the sides of a cevian triangle of some point and take their intersection points with the corresponding sides, then the resulting intersection points will lie on one straight line, called trilinear polar starting point. Orthocentric axis - trilinear polar of the orthocenter; the trilinear polar of the center of the inscribed circle is the axis of the outer bisectors. The trilinear polars of the points lying on the circumscribed conic intersect at one point (for the circumscribed circle this is the Lemoine point, for the circumscribed Steiner ellipse it is the centroid). The composition of an isogonal (or isotomic) conjugation and a trilinear polar is a duality transformation (if a point isogonally (isotomically) conjugate to a point lies on the trilinear polar of a point , then the trilinear polar of a point isogonally (isotomically) conjugate to a point lies on the trilinear polar of a point ).

Cubes

Relationships in a triangle

Note: in this section, , , are the lengths of the three sides of the triangle, and , , are the angles lying respectively opposite these three sides (opposite angles).

triangle inequality

In a non-degenerate triangle, the sum of the lengths of its two sides is greater than the length of the third side, in a degenerate one it is equal. In other words, the lengths of the sides of a triangle are related by the following inequalities:

The triangle inequality is one of the axioms of metrics.

Triangle sum of angles theorem

Sine theorem

where R is the radius of the circle circumscribed around the triangle. It follows from the theorem that if a< b < c, то α < β < γ.

Cosine theorem

Tangent theorem

Other ratios

Metric ratios in a triangle are given for:

Solving Triangles

The calculation of unknown sides and angles of a triangle, based on known ones, has historically been called "triangle solutions". In this case, the above general trigonometric theorems are used.

Area of a triangle

Special cases Notation

The following inequalities hold for the area:

Calculating the area of a triangle in space using vectors

Let the vertices of the triangle be at the points , , .

Let's introduce the area vector . The length of this vector is equal to the area of the triangle, and it is directed along the normal to the plane of the triangle:

Let , where , , are the projections of the triangle onto the coordinate planes. Wherein

and likewise

The area of the triangle is .

An alternative is to calculate the lengths of the sides (using the Pythagorean theorem) and then using the Heron formula.

Triangle theorems

Desargues theorem: if two triangles are perspective (the lines passing through the corresponding vertices of the triangles intersect at one point), then their respective sides intersect on one straight line.

Sond's theorem: if two triangles are perspective and orthologous (perpendiculars dropped from the vertices of one triangle to the sides opposite to the corresponding vertices of the triangle, and vice versa), then both orthology centers (points of intersection of these perpendiculars) and the perspective center lie on one straight line perpendicular to the perspective axis (straight line from the Desargues theorem).