Euclidean geometry


The Euclidean geometry fits all that knowledge about the design of geometric forms, being its foundation the axioms of Euclid.


What is Euclidean geometry?

Since ancient times , man developed mathematics as a way of understanding and analyzing, through symbolic language , all the phenomena that occurred in their natural environment . Among the different early applications of this field of knowledge , geometry emerged as a way of knowing and understanding in depth the shapes of bodies , highlighting the Euclidean conception of geometry above other views. The concept of Euclidean geometry It is part of the knowledge of geometric shapes taking Euclid’s axioms into consideration . It is also related to a classical view of geometry .

  • Characteristics of Euclidean geometry
  • Background
  • History
  • Postulates of Euclidean geometry
  • Elements of Euclidean geometry
  • Importance of Euclidean geometry
  • Examples

Characteristics of Euclidean geometry

Among the characteristics of Euclidean geometry are:

  • It allows the study and analysis of the plane (2 dimensions) and space (3 dimensions).
  • This form of geometry is presented axiomatically .
  • It does not recognize the existence of geometric systems where the fifth postulate does not apply.
  • It maintained notable validity in the scientific community until the appearance of Albert Einstein’s relativistic theory , in which the use of other geometric concepts became decisive .


Since the time of the caves, man has been faced with the dilemma of how to define the spatial forms for their use. In that order, cave engravings with triangles have been found that are evidence of this fact.

Among the main antecedents of Euclidean geometry are the empirical efforts made during the ancient civilizations of Egypt, Babylon , China , India, and Greece , respectively.


The geometry Euclidian was developed by Euclid of Alexandria and his disciples in the city of Alexandria , located in Egypt Old , during the reign of Ptolemy I . In this city Euclides maintained an important intellectual activity , in which he dedicated himself to the teaching of the mathematical precepts of the time and to the writing of its Elements .

In this collection of books he made an exhaustive compendium of Greek mathematics with special emphasis on proof . Among his most outstanding students can be mentioned Archimedes , among others.

Although the publication of the book was transmitted throughout the Byzantine Empire and throughout the Ancient Arab World , it was not until the middle of the late Middle Ages that it began to spread in the regions of Western Europe .

Postulates of Euclidean geometry

Among the postulates used by Euclid are the following:

  • Given the existence of two points, it is possible to draw a line such that said line joins them.
  • Any segment can be continuously extended in any direction.
  • It is possible to trace a circle by setting its center at any point and assuming any radius.
  • All right angles have the same dimension and shape without compromising their orientation.
  • Through any external point of a line it is possible to draw another unique parallel line .

Elements of Euclidean geometry

Around 300 BC . This compilation of books was written by Euclides in which he gathered all the existing knowledge regarding geometry at that time but under a deductive premise .

It is one of the most studied books in history , at the level of the Christian Bible . It also has countless editions made over time, among which is the edition made by Archimedes of Syracuse .

They are made up of 13 books, framing from 1 to 4 on conceptions on plane geometry, from 5 to 10 on proportions and reasons, to finally culminate with 3 books on geometries of solid bodies (three dimensions) .

Importance of Euclidean geometry

The importance of Euclidean geometry is that, for the first time, the geometric precepts of that time were deductively demonstrated , which represented the leap from geometry to the category of exact science .

In this way, the subsequent development of Physics as a field of study for the explanation of natural phenomena was possible . It also allowed the emergence of engineering as an applied science .


Some examples of Euclidean geometry that can be mentioned include:

  • Proposition: a triangle with equal sides has equal opposite angles of those sides.

The proof is to suppose a triangle ABC with sides AB and AC of the same magnitude. By drawing a line from point A to the middle of side BC, it is possible to obtain two triangles ABD and ACD with sides and angles of the same magnitude, which shows the equality of the angles ABD and ACD, respectively.

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