Euclidean Geometry

Euclidean geometry can be defined as the study of geometry (especially for the shapes of geometrical figures) which is attributed to the Alexandrian mathematician Euclid who has explained in his book on geometry which is known as Euclid’s Elements of Geometry. This geometry can basically universal truths, but they are not proved. Euclid introduced the geometry fundamentals like geometric figures and shapes in his book elements and has also stated 5 main axioms or postulates. We are going to discuss the definition of Euclidean geometry, Euclid’s elements of geometry,  Euclidean geometry axioms and the five important postulates of Euclidean Geometry.

What is Euclidean Geometry?

Let’s know what is Euclidean Geometry-

• Euclidean Geometry is an axiomatic system. Here all the theorems are derived from the small number of simple axioms which are known as Euclidean geometry axioms. 

• We know that the term “Geometry”  basically deals with things like points, line, angles, square, triangle, and other different shapes, the Euclidean Geometry axioms is also known as the “plane geometry”. 

• Euclidean Geometry deals with the properties and the relationship between all the things. Euclidean geometry is different from Non-Euclidean. They differ in the nature of parallel lines. 

• In Euclid geometry, for the given point and a given line, there is exactly a single line that passes through the given points in the same plane and doesn’t intersect.

Elements of Euclidean Geometry

Euclid’s Elements can generally be defined as a mathematical and geometrical work consisting of thirteen number of books that is written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Now further, the ‘Elements’ was further divided into thirteen books which had popularized geometry all over the world. As a whole, these Elements are basically a collection of definitions, postulates or axioms, propositions ( that is theorems and constructions), and mathematical proofs of the propositions.

We have already discussed what is Euclidean geometry , now let’s know what are Euclid’s axioms or Euclidean geometry axioms.

Euclid’s Axioms

Euclid Geometry Postulates:

Let us discuss a few terms that are listed by Euclid in his book 1 of the ‘Elements’ before discussing Euclid’s geometry Postulates .The postulated statements of these are as follows:

• Assume that the three steps from solids to points as solids-surface-lines-points. And now in each step, one dimension is lost.

• A solid has generally has three dimensions, the surface has two dimensions, the line has 1 and the point is dimensionless.

• A point is anything that has no part, and a breadth less length is a line and the ends of a line point.

• A surface is something which has only length and breadth.

Let’s get know these Euclid’s geometry postulates in a better way!

Questions to be Solved:

Question 1): If a point C lies between given two points A and B such that AC is equal to BC, then prove that AC is equal to \[\frac{1}{2}\] AB. Explain your understanding by drawing the figure.

Solution)Given, the length of AC = BC

(image will be uploaded soon)

Now, you need to add AC on both sides.

L.H.S + AC = R.H.S + AC

Then, AC + AC = BC + AC

We can now write 2AC = BC + AC

Since, we  already know,

BC +AC = AB (as it coincides with the given line segment AB, from figure)

Therefore, 2 AC = AB (If equals are added to equals, then the wholes are equal.)

⇒ AC = \[\frac{1}{2}\] AB.