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Geometry is among the broadest and diverse mathematical subjects. With several branches describing it, it involves the study of shapes, sizes, and features of various dimensions. In this article, we will delve deep into one of the most common type of geometry, Euclidean geometry. You can as well seek for geometry math help by getting in touch with us for prompt assistance.

What is Euclidean Geometry

Euclidean geometry is a branch of mathematics that deals with the study of plane and solid figures based on axioms and theorems employed by the Greek mathematician, Euclid. As stated earlier, it is the most common type of geometry involving the solid and plane geometry that is commonly taught in schools, especially the high school level.

Euclidean geometry requires mathematicians to have real insight into this field and try as much not to memorize simple algorithms when solving problems. In his excellent book, The Elements, Euclid only used the ruler and the compass for geometric construction which are still being used in Euclidean geometry to date.

Euclid wrote this text known as The Elements around 300 B.C, probably summarizing and summing up most of what was well known about geometry in the Greek-speaking world at the time. The book is essential for these reasons: its comprehensive contents, and it's a typical method of presentation.

For more than a millennium, the term Euclidean geometry was trivial because geometry had not yet been formulated. The Elements commence with plane geometry as the first axiomatic system. It then proceeds to the solid geometry of three dimensions.

Euclid's Axiomatic Approach

Euclidean geometry has been illustrated as an axiomatic system whereby all theorems are procured from a finite number of axioms. Euclid gives the five axioms when the book commences. The five postulates are:-

  • A straight line can be drawn joining any two points.
  • Any straight line segment can be extended indefinitely in a straight line.
  • A circle can be drawn with both the sections as the radius and an endpoint as the core given any straight line segment.
  • All right angles are congruent, are equal in other words.
  • If any two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. It is equivalent to what is also known as the parallel postulate.
  • The fifth postulate cannot be proven as a theorem, even though a number of people attempted it. Euclid only used the first four axioms for the first 28 propositions of The Elements but was forced to invoke the parallel postulate on the 29th postulate.
  • The Elements, in addition to the five postulates, includes five notions:-

  • Things which equal the same thing also equal one another.
  • If equals are added to equals, then the wholes are the same.
  • If equals are subtracted from equals, then the remainders are equal.
  • Things which coincide with one another equal one another.
  • The whole is greater than the part.
  • These conventional notions are sometimes referred to as axioms. They refer to scales of one kind. The several types of magnitudes that occur in The Elements are lines, angles, plane figures, and solid figures. Amplitudes of the same form can be compared and added, but magnitudes of different styles can neither be compared nor combined.

    The arguments of Euclid's elements also has 23 definitions, some of which are bare statements of meaning. Some of the descriptions include:-

  • A point which is a fixed location in space. Points describe a position, but have no size or shape themselves, and they are labeled using a capital letter.
  • A line which is defined as a set of infinitely many points that extend forever in both direction. They are always straight and do not take up any space and have no width.
  • The line segment is the part of a line between two points that do not extend to infinity. They can be labeled just like lines.
  • A ray is just something in between a line and a line segment: it only continues to infinity on one side. One can think of it like sunrays, they start at a point and then keep going forever, among other definitions.
  • The remaining part of The Elements consists mainly of propositions and theorems and their co-occurring proofs and constructions. Each scheme is a geometric statement, and its evidence is required to rely solely on these initial definitions, postulates, and conventional notions.

    Despite its ancientness, it persists as one of the most important theorems in mathematics. It enables us to calculate distances or, more importantly, to define ranges in situations far more general than elementary geometry.

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