The Missing Theorem
Leon O. Romain
Copyright (c) 2001 - 2002 Leon O. Romain
All rights reserved
The missing theorem establishes a simple and elegant relation between
the sides of a specific
triangle. It brings new light to known solutions of common
construction problems, establishes
specific equations for structures that were until now only
approximated, and finally challenges
the basis for certain mathematical proofs and the approach leading to
their development.
The missing theorem relates to any triangle in which one of the angles
measures twice another.
For reasons of convenience, let us call it “the Romain triangle”. For
the same reasons, we will
designate the first angle as “the double angle” and its half as “the
single angle”. Following are
the statement and proof of the said theorem.
The Romain triangle theorem
In a Romain triangle the square of the side opposite the double angle
equals the sum of the
square of the side opposed to the single angle and the product of that
side by the third one.
In more general term, the Romain triangle theorem may be expressed as
follows:
In a triangle if an angle measures twice another, the square of the
side opposite that angle is
equal to the sum of the square of the side opposite its half and the
product of that side by the
third one.
The following illustrates the Romain triangle theorem.
If ABC is a triangle in which an angle <BAC measures twice another
<ABC then: a^2 = b^2 + bc
or BC^2 = AC^2 + AC * AB
C
/\\__
/| \ \__
/ | \ \__
/ | \ \__
/---|----\--------\
A H D B
N.B.: By convention a, b and c are the sides opposed to angles A, B
and C respectively. The
caret (^) represents the exponent such that a^2 is the square of a and
a^1/2 is the square root
of a. Multiplication is represented by the asterisk (*).
Proof
Let us draw a segment CH perpendicular to the side AB. Let us also
draw another segment CD
equal to AC that meets AB in D. Then we know that angle <ADC equals
<DAC triangle ACD
being isosceles. It thus results that triangle BDC is also isosceles
since the external angle to
triangle BDC, angle <ADC, measures twice the size of one of its non
adjacent angle <ABC.
Therefore segments BD and DC are equal. Let us then compute the
relative sizes of the sides
of our original triangle.
We have:
(1) BH^2 = BC^2 - HB^2 (Pythagorean theorem)
(2) BH^2 = CD^2 - HD^2 (Pythagorean theorem)
(1)(2) BC^2 - HB^2 = CD^2 - HD^2
BC^2 - (HD + DB)^2 = CD^2 - HD^2
BC^2 - (HD^2 + 2 * HD * DB + DB^2) = CD^2 - HD^2
BC^2 - HD^2 - 2 * HD * DB - DB^2 = CD^2 - HD^2
by eliminating HD^2 we have
BC^2 - 2 * HD * DB - DB^2 = CD^2
by factorizing DB we obtain
BC^2 - DB (2 * HD + DB) = CD^2
however DB + 2 • HD = AB and CD = DB = AC therefore
BC^2 - AC * AB = AC^2
by transposing the product of AB and AC we obtain the required proof
BC^2 = AC^2 + AC * AB
or (3) a^2 = b^2 + bc
Although there is evidence that mathematicians have been familiar with
the Romain triangle for
more than two thousands (2000) years, the theorem demonstrated above
was never discovered
until now. This is why the theorem is called the missing theorem. For
more on this subject,
please visit us at: http://www.kafou.com/book.
Implications of the Romain triangle and its missing theorem.
The missing theorem is very useful in geometric constructions. It
greatly simplifies many difficult
drawing problems in geometry. For example, I will ask the reader to
try to formulate from memory
a method to build a regular pentagon or decagon. In a survey that I
have taken among advanced
mathematics students, none could come up with a valid answer. However,
a triangle formed by
two seventy two (72) degree angles and a thirty six (36) degree angle
is in fact an isoceles Romain
triangle. Therefore, using the missing theorem to compute the length
of the sides of that triangle,
the construction of these polygons becomes a very trivial task even
for a grade school student.
However, the most surprising implication of the missing theorem is
that it contradicts many facts
established by Carl Frederich Gauss and Pierre Laurent Wantzel about
two hundred years ago.
The following is a succinct expose of that contradiction.
At this point, we should note that in a Romain triangle the double and
single angles trisect the
external angle that is not adjacent to them. Two hundred years ago
Gauss stated that only the
angles that measure in degree a number equal to a Fermat prime are
constructible in Geometry
with a straightedge and a compass as tools. A short while later in
1837 Wantzel in his excellent
study of the possibility of construction with straightedge and compass
established the conditions
necessary and sufficient for any figure to be constructed with these
tools. Simply stated, since
the equation of a circle is of the second degree and that of the
straight line is of the first degree
they can only construct figures whose algebraic equation yield roots
that are a power of two. By
doing so, he put to rest many Geometric constructions that have eluded
mathematicians for many
centuries. The duplication of the cube is the most obvious
impossibility since it depends upon the
construction of a segment equal in length to the cubic root of 2. He
then established an equation
for the trisection of a sixty degree angle and stipulated that the
said equation has no constructible
roots. By doing so, he demonstrated that the trisection of a general
angle is impossible if using a
straightedge and compass as tools. Finally, in 1882 Lindemann, by
successfully proving that Pi is
a transcendental number, closed the books on the problem known as the
quadrature of the circle.
Although it is now clear that Pi can not be the root of an algebraic
equation and that the cubic
root of 2 is not a constructible number, no one is quite sure of the
real nature of the roots of the
series of equations leading to general trisection. This is because of
a shortcoming of the known
method used to obtain the roots of a cubic equation. This method was
established by a circle of
Italian mathematicians in the sixteen century and was first published
by Gerolamo Cardano in 1545.
This method was shown early on to yield complex values when in fact
the roots are real numbers.
Such is the case of the equation x^3 - 78x -220 = 0 which will yield
only complex roots when using
Cardano's method. However, it can be easily shown that 10 is a root of
that equation and that the
two other roots are also real non complex numbers. This is because in
certain cases, Cardano's
method provides results that come in the form of the cubes of two
complex conjugates whose
real part, which is the actual root, can not be determined by any
known principle. In his proof,
Wantzel used a lemma to show that the equation for the 20 degree angle
can not be reduced to
roots of the second degree. However, using the missing theorem we can
study the roots of this
series of equations in their general form and establish that their
real nature is in contradiction with
the results of Wantzel.
The nature of the roots of the equation of general trisection.
In a Romain triangle, the sides opposed to the double and single
angles form an angle that is
supplementary to the angle to be trisected. If we can deduce a general
equation between these two
sides, we can easily determine the possibility of the construction of
these triangles. In fact, the
angle to be trisected is always known and the length of one of the
sides of interest can also be
arbitrarily chosen as a constructible number. If the root of the said
equation can be proven to be
always constructible then general trisection as well as the famous
twenty degree angle can be
constructed with straightedge and compass contrary to the results of
Gauss and Wantzel.
To formulate the equation mentioned above, we must combine the missing
theorem with the
cosine law using the supplementary of the angle to be trisected. By
using the triangle of our
previous demonstration we can deduce the following equations:
(3) a^2 = b^2 + bc (Missing theorem)
(4) c^2 = a^2 + b^2 - 2ab(cosC) (cosine law)
By combining equations (3) and (4) we get
(5) a^3 - 3ab^2 + 2b^3(cosC) = 0 (See note at the end)
Applying Cardano's method to equation (5) we obtain the following
root:
(6) a = b ( -cosC + sinC(-1)^1/2 )^1/3 + b( -cosC - sinC(-1)^1/2 )^1/3
or
(6) a/b = ( -cosC + sinC(-1)^1/2 )^1/3 + ( -cosC - sinC(-1)^1/2 )^1/3
(See note at the end)
Equation (6) provides us with a single relation between the sides a
and b of the triangle. However,
we are faced with two major problems. The solution involves the sum of
two cubic roots which is
not constructible if those roots can not be reduced to powers of two.
Furthermore, both cubic roots
are complex numbers which definitely can not be constructed within the
parameters of Euclidean
Geometry. However, we know that these roots are also solutions to the
sides of Romain triangles
that are perfectly constructible such as the one formed by 90, 60 and
30 degree angles. Therefore,
we may safely say that the ratio of a and b is equal to a real non
complex quantity 2N to which the
complex value Xi is added and subtracted. The complex number Xi is
formed by a real number X
multiplied by the imaginary number i which is equal to the square root
of -1 or (-1)^1/2. We can
then write:
(7) a/b = 2N + Xi - Xi or a/b = (N + Xi) + (N - Xi)
Raising the two complex conjugates within parenthesis to the third
power we arrive at the following
relation:
(8) a/b = (-(-N^3 + 3NX^2) + (3XN^2 - X^3)i)^1/3 + (-(-N^3 + 3NX^2) -
(3XN^2 - X^3)i)^1/3
Equation (8) perfectly matches equation (6) and we may deduct the
following relations:
(9) -N^3 + 3NX^2 = cosC and (10) 3XN^2 - X^3 = sinC
(See note at the end)
Equations (7), (8), (9) and (10) clearly show that the general
equation for the trisection of any angle
always have a non complex root or that the ratio of the sides a and b
is a real number 2N.
However, to prove that these roots are constructible, we must show
that these roots can also be
reduced to powers of 2. In order to do so we must go back to the
missing theorem. We must
realize that whenever the value of side c is a real constructible
number such as the unitary segment
the equation will be expressed as follows:
(11) a^2 = b^2 + bk
With k being the real constructible value of side c. In the case where
k = 1 the equation becomes:
(12) a^2 = b^2 + b
Equation (5) being independent of side c one of its roots expressed as
a ratio of the two sides a and
b in equation (6) must be valid for all values of side c including
those mentioned above in equations
(11) and (12). Therefore, the ratio of sides a and b must be a root of
degree not higher than 2. On
the other hand, let us suppose that a few of these ratios are not
constructible such as in the case
of the 20 degree angle proposed by Wantzel. Let us use a trisectrix to
build the triangle. If we then
vary the sides of that triangle proportionally until side c becomes
equal to the unitary segment, we
end up with the case established in equation (12). This is impossible
because a quantity can not
be simultaneously constructible and not constructible.
The preceding demonstration clearly shows that the roots of the
general equation leading to the
trisection of any angle is a real constructible number. Therefore,
contrary to the universal "belief"
within the Mathematics establishment, the trisection of the general
angle is indeed possible with
a straightedge and a compass alone.
It should be noted that this expose is not a recipe for the trisection
of a general angle. It is simply
a counter proof to the results of Gauss and Wantzel concerning certain
geometric constructions.
However, it is based on sound Mathematics principles that have been
established centuries ago.
It is simply amazing that the missing theorem has never been
discovered over the past centuries.
In fact we know that during the past two thousand years, the best
minds in mathematics and other
sciences from Nicomedes to Gauss were familiar with the Romain
triangle.
For any comments, suggestions or criticism I may be reached at
le...@kafou.com.
Please forward this text to all Mathematics departments you may know
of or to any individual or
group that might be interested in these results.
Note: For a line-by-line demonstration of the proofs discussed above
you should visit the World
Wide Web address: http://www.kafou.com/book. For a complete discussion
on the subject, you
may purchase my book:
Angular Unity
The case of the missing theorem.
The book also contains many other original theorems and constructions
that are not discussed on
line. It may be purchased on the Internet at:
http://www.kafou.com/book.