An equilateral polygon has all its sides equal.
An equilateral is a figure that has all sides equal to each other . The term is usually applied to triangles of this type. An equilateral triangle , therefore, is a polygon with three identical sides , which has three acute angles equal to 60º.
It should be noted that a triangle is a polygon or figure that has three sides . These sides are made up of segments of different lines that meet at points known as vertices . Triangles meet several conditions: the sum of the length of two of their sides, for example, always exceeds the length of the remaining side.
Creating an equilateral triangle
These features (equal length sides and congruent angles) make creating an equilateral triangle simple. One way to construct an equilateral is to draw a circle with a compass , then open the compass to a measurement 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 line. You must draw a circle that has its center in X , whose radius is identical to the distance between X and Y , and a circle with its center in 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.
It cannot fail to be mentioned that human beings have learned to construct equilateral triangles in ancient times , as can be seen in several archeological sites that present figures made thousands of years ago.
But triangles are not the only polygons whose sides can measure the same. A known case is the rhombus, an equilateral quadrilateral, which includes the figure of the square .
An equilateral triangle can be drawn with the help of a compass.
Properties of these polygons
From the properties of equilateral polygons, it can be said that:
- In the case of an equilateral polygon whose angles are all of the same measure , we speak of a regular polygon.
- If an equilateral polygon is also cyclic, that is, its vertices are placed 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.
Viviani and equilateral triangles
The Italian mathematician and physicist Vincenzo Viviani developed a theorem that bears his name and which proposes that if the distances from each side of an equilateral triangle to a point are added , the result will be equal to the height of said 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 in ternary diagrams (which represent systems composed of three variables), such as those of flammability, and in simplex, which is the equivalent of a triangle in dimensions greater than 2.
Napoleon's theorem
Another well-known theorem 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, as long as the three are inside or the three outside the first, the central points of each of the new ones will form an equilateral triangle. .
The equilateral triangle in theology
For theology , the equilateral triangle has 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, explained that this geometric figure could be understood as harmony, proportion and divinity.