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Guofang Xie's Great Theorem about Conics(2)

— Perhaps the Greatest and Most Marvelous Theorem in Geometry

by Guofang Xie                         Email:  roixie@163.com   

Table of Contents

1. A Gem in Projective Geometry -- Guofang Xie's Little Theorem 

2. A Super Conjecture -- Can it Be True? How to Prove it?

3. Proof of the Super Conjecture -- Birth of a Great Theorem

 

2.  A Super Conjecture -- Can it Be True? How to Prove it?  

The above theorem enables us to construct precisely a Poncelet hexagon (i.e. a hexagon inscribed in a conic and at the same time circumscribed about another conic)  in the following way: [1]

First draw an ellipse, pick an arbitrary point inside it and draw three lines passing through this point, each of them meets the ellipse at two points, generating altogether six points (see Figure 2.1), at each of these six points draw a tangent line, the hexagon formed by the six tangent lines, which constitute its six sides, is a Poncelet hexagon (see Figure 2.2).

 Figure 2.1   Construction of a Poncelet Hexagon (step 1)

 Figure 2.2    Construction of a Poncelet Hexagon (step 2)

 

Being able to draw a precise Poncelet hexagon has tremendous value in the discovery of a much more marvelous result , in relation to which Theorem 1 (i.e. Guofang Xie's Little Theorem) is merely a harbinger, a hint , a partial revelation, like the first gemstone one digs up in a huge buried treasure .

To explore further and  "dig" deeper, One only needs to bring Poncelet’s Porism (also known as Poncelet's Closure Theorem) [2]  into play.

Having constructed a Poncelet hexagon and the two conics associated with it, by starting from any new point on the outer conic and repeatedly drawing tangent lines to the inner conic we may obtain a new Poncelet hexagon because of Poncelet’s porism, to each such hexagon we can apply Theorem 1 and find the corresponding concurrent point, it’s interesting to observe how this point changes as the vertices move around the outer ellipse.

Amazingly and thrillingly, it turns out that it is fixed! What a marvelous phenomenon! (see the figure below)

 Figure 2.3 

What is the nature of this mysterious point? What on earth causes so many lines to concur at this point? (as the vertices move around the outer ellipse, it looks as if they were “rotating” about it.)

Everyone with a little bit of intellectual curiosity will ask these questions, the reader is encouraged to stop and speculate freely , put forth possible explanations (remember Einstein's words: Imagination is more important than knowledge!), the answers will come up later in the paper.

 

At the moment the whole thing seems to be an impenetrable mystery, yet there is one sure conclusion we can draw from inspecting Figure 2.3 , it is the discernment that this mysterious concurrent point actually only depends on the two conics (although our discovery of it is made by drawing a particular hexagon between these two conics) .

Thinking along this line of thought, we soon realize that the number six (corresponding to hexagons) is not essential in this matter, if the concurrent point is only determined by the two conics, then the same thing could happen with other even-sided polygons: quadrilaterals, octagons, decagons, etc.

Precise drawing exactly confirms our conjecture ( see figures below).

 Figure 2.4 

 

Furthermore, when the two conics are two circles, we observe that the concurrent point lies on the line joining the centers of the two circles ( see figures below).


 Figure 2.5   two bicentric quadrilaterals ( O1 , O2 are the center of the inner circle and the outer circle respectively.  )

Figure 2.6   two bicentric hexagons ( O1 , O2 are the center of the outer circle and the inner circle respectively.  )



 Figure 2.7   two bicentric octagons (the two points in the middle are the center of the outer circle and the inner circle respectively.  )


 Figure 2.8   two bicentric eight point stars (the green point and yellow point in the middle are the center of the inner circle and the outer circle respectively.  )


 Figure 2.9   two bicentric ten point stars (the red point and green point in the middle are the center of the outer circle and the inner circle respectively.  )

 

Emboldened by such perfect confirmations, we are led very naturally to the following conjecture:

Super Conjecture 

If an even-sided polygon is inscribed in a conic and at the same time circumscribed about another conic, then its major diagonals and the lines joining two tangent points on each pair of opposite sides are all concurrent at a point, moreover, this point only depends on the two conics.

(Supplement) In the case when the two conics are both circles, the concurrent point lies on the line joining the centers of the two circles.

Without any doubt, this is really a breathtaking and mindboggling conjecture , once proved, it may rank as one of the greatest theorems in geometry, comparable and closely related to Poncelet’s porism. (It’s surprising that Poncelet himself did not notice it after discovering the theorem bearing his name.)

Happily and miraculously, this most daring and amazing conjecture is in fact true and can be proved, and we’re going to prove it in the later part of the paper. The author is firmly convinced of its truth before starting seriously searching for a proof, because in numerous precise drawings using high-precision geometry drawing software all the angle measurements to test collinearity turn out to be exactly 180°, accurate to one millionth of a degree(the highest precision allowed by the software, see the figure below for an example). In other sciences like physics and chemistry, this (repeated high precision experimental confirmations and the absence of counter-examples) is sufficient to establish it as a law, but in mathematics, in order to elevate it from a conjecture to a theorem , we must produce a rigorous proof.

 

 Figure 2.10   All Angle measurements confirm exact collinearity

 

But how are we going to prove it? How in the world can such a general statement about all even-sided polygons be proved? We've just seen that just proving one special case---the case of hexagons is not easy, it's a feat accomplished by the uniting force of three most powerful weapons in projective geometry: Pascal’s theorem, Brianchon’s theorem and the principle of polar reciprocity, yet all these seem powerless to tackle the general case (even in the case of hexagons, to prove that the concurrent point is fixed seems to be beyond their power).

So we must search for new methods, new ideas, and more importantly, a good reason that can explain the concurrency of so many (actually infinitely many) lines, rather than merely verify the truth of the fact, in other words, we need to find the fundamental cause of this mysterious phenomenon. Since the pattern is so uniform (the same thing seems to be happening with all even-sided polygons), we surmise that some universal principle must be at work here, finding it is finding the key to unlocking the whole mystery.

Luckily, as often happens in mathematics and in nature, such a universal underlying principle does exist, and discovering it is discovering the path to a proof. Although the actual process of proving is quite long and involves a lot of intricate technical details (herein lies the beauty and profundity of our final Super Theorem, a revelation of the wondrous unity and harmony of the world of mathematics), the main idea is extremely simple and natural, once hit upon, everything becomes clear in a flash.

 

 

 


 

 

To be continued(à suivre)

 Notes

[1] Note that Poncelet hexagon is the first Poncelet polygon whose construction poses a problem. Poncelet triangles, quadrilaterals and pentagons are all trivial and can be readily constructed, since given a generic pentagon, one can always construct a conic passing through its five vertices (such conic is unique) and another conic (which is also unique) tangent to its five sides , as for triangles and quadrilaterals, such conics are infinitely many.

[2] For an introduction to Poncelet's Porism (i.e. Poncelet's Closure Theorem), see

(1) 《伟大的彭赛列闭合定理 — The Great Poncelet's Closure Theorem》 (in Chinese)

(2)  http://mathworld.wolfram.com/PonceletsPorism.html (in English)

 




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