Equilateral triangle . In flat Euclidean geometry are those triangles that have all their sides equal.
The definition comes from the Greek where the terms x and latero refer to equals and sides respectively.
This class of triangles due to the equality of their sides has a series of properties that particularize other geometric and calculation characteristics, which have been studied since the earliest times of Humanity .
Summary
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- 1 Notable lines of the equilateral triangles.
- 2 Length and area.
- 3 Maximum area
- 3 References
- 4
Definitions.
Let be a triangle that has all its sides of the same length equal, it is said to be an equilateral triangle .
Properties.
The peculiarity of the equilateral triangles of having the 3 equal sides, makes all their geometric and calculation properties have very remarkable characteristics, with respect to other figures and other kinds of triangles.
First: every equilateral triangle is also isosceles , and can be taken as the base of any of its three sides. The reverse is not valid.
The three angles of the triangle are equal to each other and have an amplitude of 60 o . This is evident if one remembers the property that equal angles oppose equal angles in the same triangle and since the sum of the interior angles of a triangle must be 180 or , each must have an amplitude of 60 or to satisfy both properties of Euclidean geometry .
If it is known that the points A = (x A ; y A ) , B = (x B ; y B ) , C = (x C ; y C ) are the vertices of a triangle, for it to be equilateral it must be fulfilled that the distance between each pair is the same.
Equilateral triangles are the simplest regular polygons.
Given an equilateral triangle ABC and point M, the centroid, when joining points A, B, M, C, A, constructs a concave quadrilateral symmetrical with respect to bisector A, with vertex M of the entering angle; the diagonal AM inside the quadrilateral and BC, diagonal outside the polygon. [one]
Notable lines of the equilateral triangles.
Furthermore, all the notable lines of the triangles relative to each side coincide with the respective height that have the same length h as seen in the figure below:
where . This means that the perpendicular bisector, bisector, median and height coincide in the equilateral triangles and that they are also all the same, but this has other implications.
Let the points A , B , C be defined as before, the vertices of an equilateral triangle, the coordinates of the segments corresponding to each notable line would be given by a very simple method. It would be enough to remember that a median is the line that goes from one vertex of the triangle to the midpoint of the opposite side and as an equilateral triangle median it coincides with the other notable lines. So if:
They are the coordinates of the midpoints corresponding to the segments , , respectively; then each medium correspond to the segments , , .
Also the known points of convergence of the notable lines: circumcenter, barycenter, incenter and orthocenter, coincide in the case of the equilateral triangle and for us its coordinate can be found in 2/3 of the length of the segment , starting from A or expressed vectorially or its coordinates:
This same property allows determining both the radius of the inscribed circle ( r ) and that of the circumscribed ( R ) with the values:
In the case of r it can also be called for the equilateral triangles as apothem, since these are the simplest regular polygons.
Length and area.
Once the value of a (side of the equilateral triangle) is known, the determination of the perimeter 2p and the area A is given by the formulas:
- 2p = 3a.
- A = a 23 1/2 / 4
- A = 3 3/2R 2 /4, where R is the radius of the circumscribed circle.
- A = r 23 3/2 . where r is the radius of the inscribed circle. [2]
Maximum area
Among the triangles inscribed in a circle, the equilateral triangle is the one with the largest area. This results from the formula A = abc / R, with the proviso that a + b + c = 2p is constant; for a product of several factors to reach the maximum, it is necessary that the factors are equal; then a = b = c, this implies that the triangle is equilateral. [3]
The specific cases of the three trigonometric theorems (sine, cosine, tangent) take particular versions of which only that of the sines reports practical utility:
- ( Sinus theorem).
The one for cosines returns only the well-known result of cos 60 o = 0.5 and the one for tangents is undefined.
