Cooordinate Geometry 03: The Euclidean geometry

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For understanding coordinate geometry, we need to first carefully understand it's soul ~ simple middle school geometry

Who was Euclid

Euclid was a mathematics researcher based at a library in Alexandria in 325 bce to 265bce ~ same time frame as rule of Chandragupt, Bimbisara and Ashok combined. Euclid was a avid philosopher and interweaved the principles of our world beautifully in the field of Geometry. He was later also a teacher to Pythagoras.. they both viewed of geometry as an abstract model of the worldly principles.

Friends, researcher around the globe are always keen and when some human tries to establish some field of science by interweaving worldly principles into it ~ then we all recognise that as a “true art”. Take of Java programming as an example. Java beautifully interweaved objects into programming using a new class of processing called object oriented programming (OOP)... Which was opposed to procedure oriented programming. This is an example of “true art”..

Euclidean Rules at Geometry

Euclid gave various postulates of geometry.
By postulates we mean to say "worldly universal truths"
By axioms we mean to say "common assummed notions"

At first Euclid defines coordinate space. He say that there are three steps to our reality:

silence or 0 dimensions » 1dimension » 2Dimension » 3Dimension

The peak in everyday lives we could achieve is 8D as there are eight directed regions in 3D space : we called that as octants (remember the previous post - coord. 02). Above 8D is only possible only in lab... (There is no satisfactory proof to my theory)

Euclid goes to say to consider the shapes going slowly from solids to points ( ( true solids-» surfaces-» lines-» points).

In each step we lose one extension, also called a dimension. So, a solid has three dimensions, a surface has two, a line has one and a point has none.

Now, let's pointify 👉 his derived axioms, as in his axioms he added pointersto interrelate points with lines with plane surfaces with true solids.

1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.

Let's us consider examples

Question says to prove that “an equilateral triangle can be constructed on any given 'line segment'.”

Solution to this could be done using rounder. Now rounders are exceptional devices, they could draw arcs of different curvatures -means one divice, but many types of curve bends.

We draw a straight line AB, put pointed top of rounder on A and open it till B, then draw an arc above AB. Repeat the same at point B in a vice versa format : Put pointer at B and taking AB length as radius We draw another arc above the segment AB. We, thus, get a point C by the intersections of the two arcs.

This the the exceptional quality of rounders. Very nice radial arcs having radii of length AB segment