# Triangle

In Euclidean Geometry, a triangle is a plane figure, formed by three not S and the three segment S which connects them. The denomination of “triangle” is justified by the presence of three Angle S in this figure, those formed by the segments between them. The three points are the tops triangle, the three segments its sides , and the three angles its angles .

The triangle is an elementary geometrical figure, following the example Point, right or Cercle. It constitutes since Antiquity an inexhaustible reserve of properties, exercises and mathematical theorems of difficulty varied. The majority of the properties and definitions stated in this article were already stated of Euclide, approximately 300 years before Jesus-Christ, as attests some its work, “'' Éléments of geometry ''”.

For the study of the triangle in other geometries, to see Triangle (nonEuclidean geometries).

## Convention of writing

Like any polygon, one names a triangle by quoting the name of his tops, for example ABC . Here, the order does not have importance, since two unspecified tops are the ends on a side of the triangle. In general, to name the lengths on the sides, one uses the name of the top of the opposed angle, into tiny: $a = BC$, $b = AC$, $c = AB$. One names an angle by using a small letter (Greek or not) tolerates the name of the overcome top of a circumflex accent when it did not amiguïté there: $\ widehat \left\{\ alpha\right\} = \ widehat \left\{has\right\} = \ widehat \left\{has\right\} = \ widehat \left\{VAT\right\}$, $\ widehat \left\{\ beta\right\} = \ widehat \left\{B\right\} = \ widehat \left\{B\right\} = \ widehat \left\{ABC\right\}$, $\ widehat \left\{\ gamma\right\} = \ widehat \left\{C\right\} = \ widehat \left\{C\right\} = \ widehat \left\{ACB\right\}$.

If one tolerates, to reduce the notations, to confuse an angle and its measure to the statements or calculations, the correct notation is for example my ($\ widehat \left\{ABC\right\}$) =40°

## Elementary properties

A triangle can also be defined like a Polygone at three sides, or like a polygon at three tops.

After the point and the segment, the triangle is the polygonal figure simplest. It is the only one which does not have a diagonal. In space, three points define a triangle (and a plan). A contrario , if four coplanar points form a Quadrilatère, four noncoplanar points does not define a polygon, but a Tétraèdre:

In addition, any polygon can be cut out in a finished number of triangles which then form a Triangulation of this polygon. The minimal number of triangles necessary to this cutting is $n-2$, where N is the number on sides of the polygon. The study of the triangles is fundamental for that of the other polygons, for example for the demonstration of the Théorème of Pick.

### Lengths on the sides and triangular inequality

In a triangle, the length of an east coast lower or equalizes with the sum lengths on the two other sides. In other words in a triangle ABC , if one notes the three following inequalities are checked:
$BC<=BA+AC$, $AB<=AC+CB$ and $AC<=AB+BC$

This property is characteristic of the triangles. Reciprocally, being given three positive Real numbers has , B and C , if the three inequalities $a<=b+c$, $b<=a+c$ and $c<=a+b$ are checked, then there exists a triangle whose sides measure has , B and C .

To check that there exists a triangle of which the lengths on the sides are has , B and C , it is enough in practice to check only one of the three inequality, that where the longest east coast " seul". (i.e if $max \left(has, B, c\right)=a$ the only inequality to be checked is $a<=b+c$, the two others being checked.)

The case of equality of the triangular inequality makes it possible to characterize the points of a segment: M is a point of the segment if and only if AM+MB=AB.

The sum lengths on the three sides of a triangle is its perimeter .

### Summon angles

The sum of measurements of the angles of a triangle is equal to 180° or π radians.

Euclide had shown this result in its Éléments (I-32 proposal) in the following way: let us trace the parallel with the right-hand side $\left(AB\right)$ passing by $C$. Being parallel, this line and the line $\left(AB\right)$ form with the line $\left(AC\right)$ of the equal angles, coded in blue on the figure opposite (angles alternate-interns). In the same way, the angles coded in green are equal (corresponding angles). In addition, the sum of the three angles of top C is the flat angle. Thus the sum of measurements of a red angle, a green angle and a blue angle is 180° (or π radians). The sum of measurements of the angles of the triangle is thus 180°.

This property is a result of Euclidean geometry. It is not checked in general in not-Euclidean geometry.

## Typology of the triangles

When the three tops of a triangle are aligned, one speaks about triangle flattened . It is equivalent to say that an angle of the triangle is flat (it measures 180° then) or that two angles of the triangle are null (they measure 0°).

The triangles admitting two right angles (90°) and a null angle (of 0°) are qualified triangles switches of it (particular case of flattened triangle). It is a borderline case because the right angles are not correctly defined.

In all these cases, one speaks about degenerated triangles . In the continuation of this article, one supposes that the triangles are not degenerated. In the case of the degenerated triangles, of many usual properties of the triangles are false or commonplace.

### Classification according to the type of angles

As the sum of the angles of a triangle is worth 180°, a triangle cannot comprise two right angles (measuring 90°) or blunt (measuring more 90°). It thus has at least two acute angles. If the third angle is:

• right, one speaks about right-angled triangle ;
• blunt, one speaks about obtuse-angled triangle (or sometimes of blunt triangle );
• acute, one speaks about acute-angled triangle (or of acute triangle ).

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