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"CLICK" below image; or,
above blue ellipse to navigate.

Emergent Ellipse Geometry
Pythagorean Triangle Pairs (PTP)
   
  As an extension of Line FG,
draw Line AB, which is
the Major Diameter of both
the ellipse through
points A, B, D, and J;
and, the ellipse through
points A, B, E, and K.

Line AF equals Lines BG, CH, and CI.
 
 
  With Points H and I as Foci,
draw an ellipse through Point E.

Point K is on the ellipse, which
is referred to as a Vector ellipse.

All Vector Lines, such as
Lines DF and EI, are equal.
 
 
With Points F and G as Foci,
draw an ellipse through Point D.

Point J is on the ellipse, which
is referred to as a radius ellipse.
 
 
Extend the Center Line as required

Draw a Vector Line EI
that is equal to Line GH
from Point I to the Center Line.

Draw a Vector Line DF
that is equal to Line FI
from Point F to the Center Line.
 
 
  Transpose Pythagorean Triangles
FGJ and HIK as indicated.

Extend the Center Line as required.
 


 

Inscribe Circles (in-circles)
FGJ and HIK with centers at

and respective diameters LM and NO.

Bisect Lines FG and HI
with Center Lines.
 
 
  Construct a paired
Pythagorean Triangle FGJ
in accordance with base, side, and
 hypotenuse as indicated in the
below Table of Formulas.

A pair of Pythagorean Triangles
have short sides,
radius (r) and base (b),
which vary by
the Elliptical Constant (EC) = .
 
 
  Construct a Pythagorean Triangle HIK
in accordance with base, side, and
 hypotenuse as indicated in the
below Table of Formulas.

Pythagorean Triangles are triangles
with a right angle; and,
all sides that are integers.
 

 
  Pulsoid Theorem (PTm) V =   P2

 
   
  =  Elliptical Constant (EC) =   One
x   =  Integer (base 10)=b/2      

 
   
  All Vector Lines,
all
Pulse Lines,
 and all soliton Lines
are equal.
.
 
  Table of Formulas  

 
   
DF & EI   =  Vector (V)=(b/2)2 =   x2
AF & CI   =  Pulse (P)=b/2 =   x

 
   
  Pythagorean Triangles  

 
   
FGJ   =  Pythagorean radius Triangle
CF   =  soliton (s)=(b22b)/4 =   x2x
FG   =  wave (w)=b2/2b =   2(x2x)
GJ   =  radius (r)=b =   2x
FJ   =  hypotenuse (h)=b2/2b+ =   2x22x+
 
HIK   =  Pythagorean Vector Triangle
HC   =  V-soliton (Vs)=b/2 =   x
HI   =  base (b), V-wave, (Vw) =   2x
HK   =  V-radius (Vr)=b2/4 =   x2
IK   =  V-hypotenuse (Vh)=b2/4+ =   x2+

 
   
  Resoloids (in-circles)  

 
   
FGJ   =  radius-Resoloid (in-circle) =   rR
LM   =  r-diameter (rd)=b2 =   2(x)
L   =  r-radius (rr)=b/2 =   x
 
HIK   =  Vector-Resoloid (in-circle) =   VR
NO   =  VR-diameter (VRd)=b2 =   2(x)
N   =  VR-radius (VRr)=b/2 =   x

The Pythagorean Theorem, a2 + b2 = c2,
is an an incredible coincidence of
number theory, which arises from the
Emergent Ellipsoid and
the Natural source of Mathematics.

A Pythagorean Triangle Pair (PTP)
can be mapped to
every integer greater than One.

Four angles (two of the six angles are right angles)
and all six sides of a Pythagorean Triangle Pair
are of unlike values.
 
The simplest equation for
the in-diameter of a circle
inscribed within any right triangle is:


 
  An in-diameter of a right triangle
equals side "a" plus side "b"
minus the hypotenuse "c".
 

Amazingly, the diameters
of both circles inscribed within
each different Pythagorean Triangle,
of a given pair, are even integers,
 that are . . . identical to one another.

Thus, the radii of the inscribed circles
are also equal integers.
 
More amazingly,
for every Emergent Ellipsoid (EEd),
regardless of size or shape,
the inscribed circle radii
are equal to the
Pulse (P) minus the Elliptical Constant, .

 

 

"CLICK" top image; or,
below blue ellipse to navigate.

 
 



One must continuously ask:
Why? Why? Why?; and, Why? again.
And, realize that Fundamental Nature is the source of all Mathematics!
Summary Epsilon equals One Proof of One Inverse Square Law Elliptical Constant Duality of Infinity
 
Natural Function Brunardot Theorem revised Fibonacci
Sequence
Challenge to Academe Pulsoid Theorem Fundamental Intrinsic Time
 
Salient Structural Parts Universal Locus Antimatter Heaven/God/Hell Philogic Entanglement


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