Triangle

A triangle ( also outdated Triangle, Latin, triangulum ) is a polygon and a geometric figure. It is within the Euclidean geometry figure in the simplest level, which is bounded by straight lines. Its boundary lines are called the sides. Under the bonnet are three angles, the so-called interior angles span. The apex of this angle is called the vertices of the triangle. Also a generalization of the triangle concept to non-Euclidean geometries is possible.

In trigonometry, a branch of mathematics, triangles play the essential role. See in particular triangle geometry.

• 3.1 Equilateral Triangles 3.1.1 properties
• 3.1.2 formulas

• 3.3.1 Trigonometric functions of a right triangle

• 4.1 Spherical Triangles
• 4.2 Hyperbolic triangles

Classification

By side lengths:

• Equilateral triangle
• Isosceles triangle
• Irregular triangle

By angles:

• Acute triangle
• Right-angled triangle
• Obtuse triangle

Pointed and obtuse triangles are also summarized under the name scalene triangle.

The general (arbitrary) triangle

Definition and properties

A triangle is defined by three points that do not lie on a straight line. They are called vertices of the triangle. The links between any two vertices are called sides of the triangle. The triangle divides the plane into two regions, the exterior and the interior of the triangle. The angle formed by two coinciding at a vertex pages is an important parameter for the characterization of the triangle.

In geometry, the vertices of the triangle as a rule with, and referred commonly as depicted counterclockwise. The side that is opposed to a corner, will be analogous, or called. Thus, for example, is the side opposite the vertex, so connecting the points. Often, with, and instead referred to the length of each side, or. The angles are, and called; is the angle at the vertex is located at the vertex and is located at the corner

• The sum of the interior angles in a planar ( flat ) triangle is always 180 °.
• The sum of the exterior angles is 360 ° accordingly. Only an outer angle is added to the sum for each vertex. Because it is the apex angle at the two outer corners of a corner, they are always the same. The sum of all exterior angles is therefore, strictly speaking, 2 × 360 ° = 720 °.
• The total length of two sides of a triangle is always greater than the length of the third side. These relationships can be expressed in the so-called triangle inequality.

This intuitively insightful properties of plane triangles follow from the axioms of Euclidean geometry.

Excellent circles, lines, and points

Each triangle has a perimeter that is a circle passing through its three vertices. The center of the circumscribed circle is the intersection of the three perpendicular bisector; are the perpendicular straight line through the centers of the sides.

The bisectors of the three interior angles also intersect at a common point, namely the center of the inscribed circle. This touches the three sides from the inside. The three circles that touch each page from the outside and the extensions of the other two sides are called excircles of the triangle.

The center of gravity of a triangle, the common intersection of the three medians, that the respective connecting lines of the vertices to the center of the opposite side. The focus in this case signals the medians in the ratio 2:1.

The three heights, so the solders of the vertices on each opposite side, intersect at a common point, the orthocenter. By means of the heights of the area of ​​a triangle can be calculated (see triangle).

Another well-known triangle is the circle at the Feuerbach circle. It is also called nine-point circle, as he. Through the three midpoints, which runs three feet of the altitudes, and the three midpoints of the upper level sections Its center lies as the focus, the circumcenter and the orthocenter on the Eulerian straight.

Calculation of any triangle

A triangle has three sides and three interior angles. Lying three information on the size of these sites and / or angle, they yield the respective missing other pages and / or angular directions, unless there are only given three angles.

Depending on which combination of known sides and / or angle is given in detail, the result is either on or ambiguous (see adjacent figure).

Thus, the congruence deliver first of all three always clearly separable constellations, which one symbolically denoted by SSS, SWS and WSW, where S is a known side and W is a known angle.

SSW- or WSS case

The SSW or WSS case, however, is only unique if the known angle of the larger of the two given sides facing (SSW - case ) - he is the smaller side compared to (SSW - case ), there are usually two different triangles, which satisfy the initial conditions. However, this must be a not always so, as the special case with an aspect ratio of 1: 2 and the angle is 30 °, which is exactly then however is only one such particular triangle, the angle opposite the longer side is 90 °. To mention a value is finally mathematically possible situation that no triangle meets the initial conditions, namely when for the sine of the longer side opposite angle > 1 results (with real existing triangles, however, this case is of course excluded ).

WWS- or SWW- case

The WWS or SWW- case can be solved (such as the adjacent figure ) to draw in two ways: either calculated by the sine theorem, first of all, one of the two remaining sides and then expects further than in the SSW- case, or is determined, which is much more convenient means of the sum of angles in a triangle the missing third angle and then Proceeding as in the SW case.

WWW case

The WWW case with flat triangles not uniquely solvable, because in this case, in fact, exist only two independent data, the size of the third angle, however, always inevitably results from the size of the other two. Without a given page, although the shape of this triangle is given, but its size remains undetermined.

Of sines and cosines

The most important tools for the calculation of any triangle are in addition to the sum of the angles in the triangle of the sine and cosine of faced by the other triangle sets as the projection theorem and tangent set and the half-angle rates play only a minor role.

The computationally most expensive, but also most powerful of the three tools is the law of cosines, as you calculate with him as the sole of a triangle without all angles are in a first angle (and then help with the simpler sinus rate and the sum of angles in a triangle ) can. Accordingly, the law of cosines is used in the context discussed here, only at the beginning of the calculation of a triangle of type SSS or SWS, while everything will rest easier and done faster by sines and angle sum.

As seen below, the law of cosines as the Pythagorean theorem starts the same way and in fact, one can interpret this as a special case of the cosine law:

If in fact the two sides of a triangle given included angle is a right, so be cosine is equal to zero, and then what is left of the cosine concerned, is nothing more than another version of " Pythagoras ".

If you know of a triangle only its three sides, and, its interior angles have the aid of the arc cosine function ( arccos ) determined as follows:

The law of sines is available in three variants, which can be summarized as follows:

As can be seen, the law of sines is computationally much more straightforward: If we know one of the three breaks, one automatically knows so well all the rest. For this, however here is always at least one of the three interior angles already be known, and if not, first of all recourse to the law of cosines (see above).

Special triangles

Equilateral triangles

Properties

A triangle in which all three sides are equal is called equilateral triangle. This is a special isosceles triangle whose three interior angles are all the same size and therefore be 60 °. This includes the equilateral triangles to regular polygons.

All equilateral triangles are similar to each other and congruent if and only if their side lengths are equal. Midperpendicular, Seitenhalbierende and height to one side and bisects the opposite angle fall in an equilateral triangle, each successive. The same applies to the circumcenter, the incenter, the center of gravity and the orthocenter of an equilateral triangle, so that this point is often simply called the center.

Formulas

For an equilateral triangle with sides of length applies:

Proof see Related links below.

Isosceles triangles

An isosceles triangle is a triangle with at least two sides are of equal length. These sites are referred to as legs, the third side is the base of the isosceles triangle. The two angles at the base ( the base angle) are equal. The point meet at the both legs is called peak, the local angle is the angle at the top. The equilateral triangle is a special isosceles triangle in which each side is base and legs simultaneously. A protractor is a right isosceles triangle.

In an isosceles triangle, the perpendicular bisector of the base, the Seitenhalbierende the base and the height to the base and the bisector of the apex angle fall onto each other. You can see the length of this path, ie in particular the amount determined, by applying the Pythagorean theorem to one half of the triangle. This results in, and thus the surface.

Right-angled triangles

A right-angled triangle is a triangle, that is has an angle of 90 ° a right angle. The right angle opposite side is the longest side of the triangle and the hypotenuse is called. The other two sides are called catheti. With respect to one of the acute angle of the triangle is defined as the angle fitting cathetus as the adjacent side and the angle opposite cathetus as opposite leg.

The lengths of the three sides of a right triangle are brought by the Pythagorean theorem in relationship: The square of the length of the hypotenuse ( the graph referred to as ) is equal to the sum of the squares of the lengths of the other sides (and). Conversely, a triangle, wherein the side lengths of each other in the relationship, a right triangle with a hypotenuse.

Trigonometric functions in the right-angled triangle

The ratio between short sides and hypotenuse Also the two acute angles of the right triangle can be uniquely determined. The following six functions are called trigonometric functions or trigonometric functions.

From the above can be represented by the reciprocal of Education the following.

The inverse functions of the trigonometric functions are called arcsine, arccosine, arctangent, etc. called - their main application is accordingly to provide to the given sine, cosine or tangent values, the corresponding angle.

Irregular triangles

• All three sides have different lengths.
• All three angles are different.

Triangles of non-Euclidean geometry

Spherical triangles

→ Main article: ball triangle

Triangles on a sphere whose three sides are parts of great circles, is called spherical triangles and spherical triangles. Your side lengths are not specified in the dimension of length ( meters, centimeters, etc. ), but as associated angle in the center of the sphere.

A spherical triangle has a sum of angles greater than 180 °. The " surplus" is called spherical excess and usually referred to in formulas:

The maximum excess of 360 ° occurs in a " triangle " with three to 180 ° stretched angles. This to a large circle degenerate triangle has the sum of angles 540 ° ( three times 180 ° ) and ε = 540 ° - 180 ° = 360 °.

The excess is directly related to the area of ​​the triangle together:

Where the radius of the sphere and the circle number 3.14159 ... means.

Spherical triangles can be calculated analogously to the flat triangles, for which there is the spherical law of sines, law of cosines to the projection set and various half- angle sets, for example, in geodesy - see Spherical Trigonometry.

Hyperbolic triangles

For non-Euclidean geometry - in which the parallel axiom does not apply - include triangles on a saddle surface. While a ball is everywhere convex curved, saddle and other hyperbolic surfaces, both convex and concave curvature have ( their product, the curvature is negative ).

Accordingly, the excess is negative - i.e. the sum of the angles of a triangle on a saddle surface is less than 180 °. The congruence theorems make statements about the triangle sizes (side length, angle), which are necessary to determine a triangle uniquely.

Sets around the triangle

• Similarity sets
• Congruence
• Pythagorean Theorem
• Set of Heron
• Theorem of Thales
• Set of Stewart
• Set of Routh
• Circles at the triangle area, inscribed circle, excircles, Feuerbach circle
• Just Euler
• Just Simsonsche
• Symmedianen and Lemoinepunkt
• Fermat point
• Höhenfußpunktdreieck
• Morley triangle
• Napoleon Triangle and Napoleon - point
• Inequality Pedoe
• Formulary trigonometry

Triangle as a symbol

The triangle is used as a symbol, as in theology, as an ideological symbol, as a mathematical symbol and also in signs.