An Approach to the Triangle

In this Section , we shall develop a { Shape Classification System } for the Triangle ,

• based on a scheme utilizing 'true' & 'pseudo' { Fermat Points } .

And this classification system does incorporate the special case of the { 3-colinear-points Triangle } :

• which is always essential & vital in solving problems such as the { Three Body Problem } .

This then lays the foundation works for further explorations & analysis in :

• Section VI --- An Infinite Series of { Triangles } , &
• Section VIII --- Mapping { Triangles } onto { Triangles } .

Let us start-off the { Shape Classification System } by introducing :

• a { Reference Circle of radius } , and

• an equilateral triangle , { Triangle R-S-T } , so that :
  • { Point R } , { Point S } & { Point T } are on the edge of the circle ,

  as per the diagram on-the-left , below :

We then identify { Point O } as the center of the circle , which is also centroid of the equilateral triangle , as per the diagram on-the-right , above .

We also identify , for reference purposes , the { 'Polar North' Reference Direction } arising from ( Point O } :

• which is always perpendicular to { Side R-S } ,

  as per the diagram on-the-right , above .

Let us now pick a point , { Point P } , which always resides within the boundaries of the circle :

We note here that :

• whenever { Point P } resides within the boundaries of the { equilateral triangle } as marked in { white-color } in the diagram on-the-left , below :
  • the point { Point P } is the { Fermat Point } for { Triangle A-B-C } ,

    because { Line A-P } , { Line B-P } & { Line C-P } are each perpendicular to a different side of an equilateral triangle and therefore intesects one-another at { 120 degrees } ,

  

• whenever { Point P } resides outside the boundaries of the { equilateral triagle } , such as in the { light-green-color } region as shown in the diagram on-the-right , above :
  • the point { Point P } is a 'pseudo' { Fermat Point } for { Triangle A-B-C } .

It is 'pseudo' in the sense that , for a triangle { Triangle A-B-C } with an { obtuse angle greater than 120 degrees } at { Point A } :

• when we apply the traditional { Fermat Point Construction Method } to it , i.e. :
  • constructing 3 equilateral triangles attached to the 3 sides and joining the opposite vertices ,

  we do arrive at { Point P } as being the intersection of { Line A-D } , { Line B-E } & { Line C-F } , as per the diagram on-the-right , below :

And we now bring-in the diagram on-the-left , above , for the 'true' { Fermat Point } , for comparison purposes .

We simply note here that :

• for the diagram on-the-left , we have :
  • { length of Line A-D } = { length of Line D-P } + { length of Line P-A }

• for the diagram on-the-right , we have :
  • { length of Line A-D } = { length of Line D-P } - { length of Line P-A }

In certain sense , the { length of Line P-A } may be consindered 'negative' when the [ angle at { Point A } ] is greater than [ 120 degrees ] .

But this does not really have any significant impact on our { Shape Classification System } here ,

• although it does lay the groundwork for a better understanding of the { Fermat Point Construction Method } ,

  to be further discussed in [ Topic #4 ] in the next section , Section VI .

Let us now look at the differently shaped triangles associated with differently located { Point P's } .

Let us first move-in from the { edge of the circle } towards the { center of the circle } :

• via the six (6) directions { Direction 000 } / { Direction 060 } / { Direction 120 } / { Direction 180 } / { Direction 240 } / { Direction 300 } ,

  as identified in this diagram below :

  

We then have these 6 tables of { triangular shapes } below :

For { Direction 000 } :

Edge of Circle	--->	Half-way	--->	Center of Circle

For { Direction 060 } :

Edge of Circle	--->	Half-way	--->	Center of Circle

For { Direction 120 } :

Edge of Circle	--->	Half-way	--->	Center of Circle

For { Direction 180 } :

Edge of Circle	--->	Half-way	--->	Center of Circle

For { Direction 240 } :

Edge of Circle	--->	Half-way	--->	Center of Circle

For { Direction 300 } :

Edge of Circle	--->	Half-way	--->	Center of Circle

We now make 2 observations :

• { Point P } for the { isoceles triangles } are always resident on these 6 afore-mentioned directions :
  • { Direction 000 } / { Direction 060 } / { Direction 120 } / { Direction 180 } / { Direction 240 } / { Direction 300 } .

• { Point P } for the { 3-colinear-points triangles } are always resident at the edge of the { circle } .

Let us now take a second look and move-in from the { edge of the circle } towards the { center of the circle } :

• via the five (5) directions { Direction 000 } / { Direction 015 } / { Direction 030 } / { Direction 045 } / { Direction 060 } ,

  as identified in this diagram below :

  

We then put these 5 Directions on a single table , for easier understanding & comparison purposes :

Edge of Circle	--->	Half-way	--->	Center of Circle

We observe , again , that :

• { Point P } for the { 3-colinear-points triangles } are always resident at the edge of the { circle } .

The reader should also take note of the { Triple-Pairs / Hexa-symmetric } nature of this system , as per this demonstration below :

Demonstration of { Triple-Pairs / Hexa-symmetry } :

Half-Way at Direction 045	Half-Way at Direction 075	Half-Way at Direction 165	Half-Way at Direction 195	Half-Way at Direction 285	Half-Way at Direction 315

• the 'absolute' shape of the 6 triangles here are either { the same } or { the mirror-image of one-another } .

And the { Triple-Pairs } consolidate into { Triple-Singles } for the { isoceles triangles } , as per this demonstration below :

Demonstration of { Triple-Singles } for { isoceles triangles } :

Half-Way at Direction 060	Half-Way at Direction 180	Half-Way at Direction 300

• again the 'absolute' shape of the 3 triangles here are always { the same } .

We hope we have developed a powerful { Shape Classification System } which may have many applications .

In our { Shape Classification System } here , { radius } of the { Reference Circle } is controlling on the size of the resultant { triangle } , i.e. :

• if we hold { Angle } & the [ ratio of { distance O-P } : { radius } ] constant ,

  the size of { Triangle A-B-C } will increase as we increase { radius } .

Our proposal on { size } here is to set up a { 3-Dimensional Space } so that :

• the { Reference Circle of radius } always lie on the horizontal plane ,
  • with the horizontal plane as marked by the { X-axis } & { Y-axis } directions in the diagram below .

• { radius } of the { Reference Circle } increases linearly as we move-up in the vertical direction ,
  • as marked by the { Z-axis } direction in the diagram below .

Any point { Point P } resident within the boundaries of the { Conic Section } would then represent a { differently sized & differently shaped triangle } , { Triangle A-B-C } .

This can be rather convenient when we investigate [ mapping { Triangles } onto { Triangles } ] in Section VIII .

But 2 preliminary thoughts here for further considerations , at this juncture :

• mapping { closed-paths } on { closed-paths } within the different { Conic Sections } ,

• ricochet off-the-conic-section-wall / off-the-edge-of-the-circle , as the case may be :
  • same as { the angle-of-incident equal to the angle-of-reflection } in { light reflection off-a-mirror } ???

Could be interesting !