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 Emergent Ellipse Geometry Pythagorean Triangle Pairs (PTP)
 Inscribe Circles (in-circles)FGJ and HIK with centers at and respective diameters LM and NO. Bisect Lines FG and HIwith Center Lines.
 Construct a pairedPythagorean Triangle FGJin accordance with base, side, and hypotenuse as indicated in thebelow Table of Formulas. A pair of Pythagorean Triangleshave short sides,radius (r) and base (b),which vary bythe Elliptical Constant (EC) = .
 Construct a Pythagorean Triangle HIKin accordance with base, side, and hypotenuse as indicated in thebelow Table of Formulas. Pythagorean Triangles are triangleswith a right angle; and,all sides that are integers.
 = Elliptical Constant (EC) = One
 x = Integer (base 10)=b/2
 Table of Formulas
 Pythagorean Triangles
 FG = wave (w)=b2/2–b = 2(x2–x)
 GJ = radius (r)=b– = 2x–
 FJ = hypotenuse (h)=b2/2–b+ = 2x2–2x+
 HIK = Pythagorean Vector Triangle
 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)=b–2 = 2(x–)
 L = r-radius (rr)=b/2– = x–
 HIK = Vector-Resoloid (in-circle) = VR
 NO = VR-diameter (VRd)=b–2 = 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 Pairare of unlike values.
 The simplest equation forthe in-diameter of a circleinscribed within any right triangle is:
 An in-diameter of a right triangleequals side "a" plus side "b"minus the hypotenuse "c".
 Amazingly, the diametersof both circles inscribed withineach different Pythagorean Triangle,of a given pair, are even integers, that are . . . identical to one another. Thus, the radii of the inscribed circlesare also equal integers.
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 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 revised Fibonacci Sequence
 Natural Function Brunardot Theorem Duality of Infinity Challenge to Academe Pulsoid Theorem Fundamental Intrinsic Time
 Salient Structural Parts Universal Locus Heisenberg Uncertainty Principle Heaven/God/Hell Philogic Entanglement