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 (
* O*_{1}
, * O*_{2}
are the center of the inner
circle and the outer circle
respectively. )

Figure 2.6 two
bicentric hexagons ( * O*_{1}
, * O*_{2}
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）

[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.