A **triangle** is a polygon or **figure** that has **three sides**. These sides are made up of **segments** of different **straight lines** found in the **points** known as **vertices**. Triangles meet several conditions: the sum of what two of their sides measure, for example, always exceeds the length of the remaining side.

A **equilateral** is a figure that presents **all sides equal to each other**. The term is usually applied to **triangles** of this type. A **equilateral triangle**, therefore, it is a **polygon with three identical sides**, which has three acute angles equal to 60º.

These characteristics (sides of equal **length** and congruent angles) make creating an equilateral triangle simple. One way to construct an equilateral is by drawing a circle with a **compass**, then open the compass by a measure of 60º and mark three equidistant points. By joining the three points, the equilateral triangle is formed.

Another option is to link a point **X** and a point **Y** through a straight line. You have to draw a circumference that has its **center** on **X**, whose radius is identical to the distance between **X** and **Y**, and a circumference with its center at **Y** and radius identical to the distance between **X** and **Y**. By joining the point where both circles intersect with **X** and **Y**, a new equilateral triangle is created.

But triangles are not the only polygons whose sides can measure the same. A known case is the rhombus, an equilateral quadrilateral, where the figure of the square is included. Among the properties that this type of polygons present, it is said that:

***** in the case of an equilateral polygon whose angles are all the same **measure**We are talking about a regular polygon;

***** If an equilateral polygon is also cyclic, that is, its vertices are perched on a circle, it will also be a regular polygon;

***** any equilateral quadrilateral is convex, although this is no longer true for polygons that exceed four sides.

The mathematician and **physical** Italian Vincenzo Viviani developed a theorem that bears his name and that proposes that if the distances from each side of an equilateral triangle are added to a point, the result will be equal to the height of that figure. Viviani’s theorem can also be proven with equilateral and equiangular polygons. One of its applications in the real world is its use to **plot coordinates on ternary diagrams** (representing systems composed of three variables), such as flammability, and in simplex, which is the equivalent of a triangle with dimensions greater than 2.

Other **theorem** known in the field of geometry is that of Napoleon, whose authorship *cannot be assured that it belongs to Bonaparte*. In its statement, it is explained that when constructing three equilateral triangles based on the sides of a triangle of any type, provided that all three are inside or all three outside the first, the central points of each of the new ones will form an equilateral triangle .

Humans have learned to construct equilateral triangles in **remote times**, as can be seen in several **deposits** of archeology that present figures made thousands of years ago.

For the **theology**, the equilateral triangle is of great importance. In principle, the number three symbolizes spiritual order, balance. According to some religious representations, the Catholic god is depicted as an inverted triangle with an eye inside it, alluding to his omnipresence and omniscience. Plato, on the other hand, stated that this geometric figure could be understood as harmony, proportion and divinity.

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