Random Triangle

On a Quick Expansion Method for Pythagorus Triples

Section VI - Setting up the Random Triangle and its Reference Frame

In this and the next 2 Sections , we shall repeat the procedure for :
- solving for the 2 mirror-imaged Equilateral Triangles arising from any given { Triangle A-B-C } ;
so that :
- in the Section thereafter , we can give full details as to :
  - how we came upon the Quick Expansion Method for Pythagorus Triples .
In this 1st of the 3 Sections ,
- we shall set up the random triangle { Triangle A-B-C } and its Reference Frame ;
  - for our further use in the next 2 Sections .

For our discussion purposes throughout this 3 Sections ,
- we shall first set up three (3) random points on a { 2-Dimensional Plane } ,
  - namely [ Point A ] , [ Point B ] and [ Point C ] respectively ;
  so that :
  - the 3 points are separate from one-another and do not lie on the same straight-line .
As such , the { 3-points shape } formed by [ Point A ] , [ Point B ] and [ Point C ] is a triangle ,
- and this { 3-points shape } shall be referred to as { Triangle A-B-C } throughout the 3 Sections here .
We can then identify [ Point O ] as the centroid of { Triangle A-B-C } and :
- set up the 3 position vectors [ Vector O-A ] , [ Vector O-B ] , and [ Vector O-C ] respectively ;
  - as per the set of 3 diagrams above .
And because [ Point O ] is the centroid of { Triangle A-B-C } , we always have :
- [ Vector O-A ] + [ Vector O-B ] + [ Vector O-C ] = [ zero ]

For our analysis to follow , it is necessary for us to set up a Reference Frame and quantify the distances .

And we shall do so now :
- FIRST ,
  - we set up a random straight-line passing thru [ Point O ] to act as the [ X-Axis ] ;
- SECONDLY ,
  - we set up another straight-line passing thru [ Point O ] orthogonal to the [ X-Axis ] to act as the [ Y-Axis ] ;
  as per the diagram on-the-left below .
We can then draw in a circle of random radius , using [ Point O ] as the center of the circle ,
- to act as our { unit-circle } for referencing distances ;
so that :
- the [ distance O-A ] , [ distance O-B ] , and [ distance O-C ] can now be quantified accordingly ;
  - as per the 2 diagrams in-the-middle and on-the-left above .