RATIONAL TRIGONOMETRY

Rational trigonometry
JUNI 2012
Triangles are one of the most basic objects in mathematics. We have been studying them for
thousands of years, and the study of triangles, Trigonometry, is, to some extent, a part of every
mathematical curriculum. Our oldest named theorem, the Pythagorean theorem, is about triangles,
though the theorem was known long before Pythagoras. It is probably our most famous and most often
proved theorem as well. Hundreds of different proofs are known, [Loomis 1940] and good writers still
find interesting things to say about the theorem. [Maor 2007]
The particular branch of trigonometry where we ask that certain parts of the given triangle, sides,
angles, medians, area, etc., is called rational trigonometry. Though it originally arose from geometry,
rational trigonometry is now usually classified as a part of number theory.
For example, for many people, the Pythagorean theorem is particularly interesting when we
consider it as a problem in rational trigonometry and ask that the lengths of the sides of the triangle be
whole numbers. This is the problem of so-called Pythagorean triples, three whole numbers a, b and c
satisfying
a2 + b2 = c2 .
As we all know, the simplest such triple is (3,4,5). It is easy to show that there are infinitely
many such triples. We can generate all we want by picking two positive integers, m and n, with m>n
and letting
a = 2mn
b = m2 ! n2
c = m2 + n2 .
It is easy to check that for these values, indeed, a2 + b2 = c2 . It is slightly less easy to check that
if m and n are relatively prime, one odd and the other even, then a, b and c are pairwise relatively prime,
so the method is not just generating infinitely many triangles similar to each other. All Pythagorean
triples can be generated in this way.
Another way to generate Pythagorean triples is apparently due to Ozanam. He tells us to look at
the sequence of rational numbers
2
11
3 , 2 2
5 , 3 3
7 , …, n n
2n+1 ,….
Each of these numbers, written as an improper fraction, a
b
, gives two of the three numbers of a
Pythagorean triple. We leave it to the reader to find why this is true.
Fibonacci also showed a way to find infinitely many different Pythagorean triples, but neither
Fibonacci's nor Ozanam's method gives all of them.
It should be no surprise that Euler also worked in rational trigonometry. He wrote about half a
dozen papers on the subject, and our purpose in this column is to look at a sequence of four of them,
giving better and better solutions to the same problem. The first of those papers [E451] gives the
problem right in its title, Solutio problematis de inveniendo triangulo, in quo rectae ex singulis angulis
latera opposita bisecantes sint rationales, "Solution of the problem of finding a triangle in which the
lengths of the straight lines drawn from each angle and bisecting the opposite sides are rational." Euler
neglects to mention that he means the sides of the triangle to be rational as well, nor that he means to
multiply by the least common denominator and make all these measures integers instead of rational
numbers. Euler wrote this paper in 1773.
The other three papers, with their titles in English and the years that Euler wrote them, are
E713 (1778) Investigation of a triangle in which the distance from the angles to its center of
gravity is rationally expressed
E732 (1779) An easier solution to the Diophontine problem about triangles, in which the straight
lines from the angles to the midpoints of the opposite sides are rationally expressed
E754 (1782) A problem in geometry solved by Diophantine analysis
The last of these was written in French. The others were in Latin, though the second one, E713,
has a short summary in French, which we quote below:
This article, which will give pleasure to the small number of amateurs in
indeterminate analysis, contains a very beautiful solution to the problem stated in the
title. Here it is in just a few words. Let the sides of the desired triangle be 2a, 2b, 2c,
and let the straight lines be drawn from their midpoints to the opposite angles,
respectively f, g, h. Take as you please any two numbers q and r and find
M =
5qq ! rr
4qq
and N =
5rr ! 9qq
4rr
. Reduce the fraction (M ! N)2
! 4
4(M + N)
to its lowest
terms, and name the numerator x and the denominator y. Then you will have the side
2a = 2qx + (M ! N)qy and the line f = rx ! 1
2 (M ! N)ry . Make p = x + y and s = x –
y, and you will have the sides 2b = pr ! qs and 2c = pr + qs and the lines
g =
3pq + rs
2
and h =
3pq ! rs
2
.
The summary shows that the spirit of Euler's solution is like that of the formulas above that give
all the Pythagorean triples. We get to choose two numbers, here q and r, with a few restrictions (like we
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don't want M + N = 0, as stated in the text but not the summary.) Then the formulas give the solutions
in terms of p and q. As with the solution to the problem of the Pythagorean triples, it is easy to see that
all the values, a, b, c, f, g and h, are indeed rational. It is a bit more subtle and a good deal more tedious
to check that these values f, g and h are the medians of the triangle with sides 2a, 2b and 2c. Some of
that will be evident from what follows.
Note that Euler mentions that this paper "will give pleasure to the small number of amateurs in
indeterminate analysis." To Euler, "indeterminate analysis" is the practice of finding integer or rational
solutions to algebraic equations, what we now call and Euler himself would later call Diophantine
analysis. He also mentions that he doesn't think that very many people will be interested, that there are
only a "small number of amateurs." I think he uses the word "amateurs" a bit differently than we use the
same word today. Now it means "people who are not professionals," but to Euler it meant "people who
love the subject." I hope we're all "amateurs" in Euler's sense of the word.
We've seen Euler's beginning, the statement of the problem, and one of his answers. Let's look a
bit at his solutions, at some of the things he discovered along the way, and at why he felt the need to
return to the problem to improve his solution.
Euler begins the first of his papers, E451 with only the title as preamble and tells us that we
should let the sides of the desired triangle be 2a, 2b and 2c and the lengths of the medians be f, g and h.
Then we want to find rational solutions to the system of equations
2bb + 2cc ! aa = ff
2cc + 2aa ! bb = gg
2aa + 2bb ! cc = hh
Euler calls these three equations his
"fundamental equations" for this problem.
He doesn't tell us here why these
equations have anything to do with the
problem, but in the second of the four
papers, E713, perhaps he is being a bit
more gentle on his "amateurs," for he gives
us details and a diagram.
Let ABC be a triangle, with midpoints F, G
and H opposite A, B and C, respectively,
and medians AF, BG, and CH intersecting at O, the center of gravity. Let a=BF=CF, b=CG=AG,
c=AH=BH, f=AF, g=BG, h=CH and ω=∠AFB.
Euler claims, without explicitly mentioning the Law of Cosines, that
AB2 = AF2 + BF2 ! 2AF " BF cos#
and
AC2 = AF2 + CF2 + 2AF !CFcos".
Add these to get
AB2 + AC2 = 2AF2 + 2BF2
or
4
4cc + 4bb = 2ff + 2aa,
or
ff = 2cc + 2bb – aa,.
Similarly,
gg = 2aa + 2cc – bb and hh = 2aa + 2bb – cc.
Thus the problem becomes to find three numbers, a, b, c, for which these three formulae produce
squares.
We'll return to E451 and follow Euler on a short tangent. If we use the three fundamental
equations, we find that
2gg + 2hh ! ff = 9aa,
2hh + 2 ff ! gg = 9bb,
2 ff + 2gg ! hh = 9cc.
In the fourth of these papers, E754, Euler describes these equations as "a pleasant property," but
that this property "does not contribute in any manner to the solution of the problem." But what is
"pleasant" about these equations. They are the same as his three fundamental equations, but with f, g
and h substituted for a, b and c, and with 3a, 3b and 3c substituted for f, g and h.
This means that if a triangle with sides 2a, 2b and 2c has medians of length f, g and h, then a
triangle with sides 2f, 2g and 2h has medians of length 3a, 3b and 3c. If the measures in one triangle are
all rational, then so are the measures in the other, and so we learn that solutions to this problem in
rational trigonometry come in pairs.
But we still don't have any solutions. All of Euler's solutions are rather long, so we will only
summarize them
In his first solution, the one given in E451, Euler rewrites his first two fundamental equations as
ff = (b ! c)2
+ (b ! c)2
! aa = (b ! c)2
+ (b + c + a)(b + c ! a)
gg = (a ! c)2
+ (a + c)2
! bb = (a ! c)2
+ (a + c + b)(a + c ! b).
Being a genius at substitution, Euler introduces two new variables, p and q, that enable him to take
square roots of these two equations and write them as
f = b ! c + (b + c + a) p
g = a ! c + (a + c + b)q.
After two pages of dense calculations, Euler finds a sixth degree polynomial that gives hh in terms of p
and q. Then he finds rational expressions for a, b and cˆin terms of p and q. Hence, if p and q are
rational, then so are a, b, c, f and g. That leaves h. So, all Euler has to do is find some rational values of
p an q that make his sixth degree polynomial into a perfect square, and at the same time, don't make any
of the denominators of his rational expressions equal to zero. It is tedious, but he manages to find
several solutions, among which are
5
1. a = 158 b = 127 c = 131 f = 204 g = 261 h = 255
and its companion solution, reduced to lowest terms because f, g and h are all multiples of 3,
2. a = 68 b = 87 c = 85 f = 158 g = 127 h = 131.
Five years later, in 1778, Euler made his second attack on the problem, E713. As we mentiond
above, the Summary of this article mentions the "Amateurs of analysis," and he uses the law of cosines
to justify his fundamental equations.
During those five years, Euler apparently realized that the "pleasant property" was not just an
interesting property of triangles with rational medians, but is a general property of all triangles. He calls
it a "most distinguished property" and states it more geometrically than he did before, writing
AO2 + BO2 + CO2 =
1
3
BC2 + AC2 + AB2 ( ).
His other calculations are quite similar, but when it comes time to introduce the new variables p
and q, he defines them as
f = b + c +
p
q
(b ! c + a).
This, combined with the first fundamental equation, allows Euler to write a and f, and hence g and h, in
terms of b, c, p and q. Things get complicated, but after a while he introduces two more variables, r and
s, to make c + b = pr and c – b = qs, and then two more, x and y such that p = x + y and q = x – y, then t
and u so that a
q
= x + ty and f
r
= x + uy , and finally M and N so that 2tx + tty = y + 2Mx and
2ux + uuy = y + 2Nx. In this tower of substitutions, everything ends up depending on q and r, and Euler
can find some triangles. We've skipped five pages of details here. The interested reader is encouraged
to consult the original sources. The mathematics there is considerably more difficult than the Latin.
In the end, Euler finds that for q = 1 and r = 2, as well as for q = 2, r = 3, he gets the same
triangle we labeled 1 above, but for q = 2, r = 1, he finds
3. a = 404 b = 377 c = 619 f = 942 g = 975 h= 477.
Then for q = 1 and r = 3, he gets
4. a = 3 b = 1 c = 2 f = 1 g = 5 h = 4.
Though this is a solution to the Diophantine equations, the sides 3, 1 and 2 do not form a triangle. He
gives several other solutions as well.
Hence, this solution lacks two of the properties we admire in the solution to the problem of
Pythagorean triples. Two different choices of the variables p and q can give the same solution, and
some choices of p and q can give inadmissible solutions. Euler doesn't seem to ask whether or not all
rational triangles with rational medians are generated in this way.
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Euler's third solution to the problem of rational medinas, E732, followed just a year later, in
1779. For this paper, he called the midpoints of the sides X, Y, and Z instead of F, G and H, and the
corresponding lengths of the medians are x, y and z instead of f, g and h. Perhaps this is a symptom of
Euler's blindness, as he had been almost entirely blind since unsuccessful cataract surgery in 1773, and
he was unable to consult his earlier works on the subject to make his notation consistent.
Using his new notation, Euler transforms two of his three fundamental equations into different
forms:
I. xx ! yy = 3(bb ! aa),
II, xx + yy = 4cc + aa + bb, and
III. zz = 2aa + 2bb ! cc.
These equations are enough different from the others that after Euler makes another sequence of
miraculous substitutions, introducing f and g, p and q, m and n, t and u, and finally M, he gets everything
in terms of f and g. This takes him just three pages of calculations, and the solution is essentially the
same as the one we translated above from E754. A few highlights are
m =
5gg ! ff
4gg
and n =
5 ff ! gg
4 ff
,
exactly as M and N will depend on r and q in E754. Likewise,
p = 4(m + n) and q = (m ! n)2
! 4,
almost like his variables x and y are defined in E754, but there he factors out their greatest common
divisor.
Now, in terms of f, g, p and q, Euler tells us that the sides of the triangle are 2a, 2b and 2c, where
a, b and c are given by
a = ( f ! g) p + ( f + g)q
b = ( f + g) p + ( f ! g)q
c = 2g(m ! n)(3m + n) ! 8g
= g(m ! n) p + 2gq.
He also gives equations for the lengths of the medians, x, y and z.
For his first example, he takes f = 2, g = 1 to get his first solution again, then f = 1, g = 2 to get
the third one. There are no new rational triangles in this paper. Its main improvement over its
predecessor, E713, seems to be that its calculations are a bit shorter, and its answer is more concise.
The last of the four papers is much like the third one. Euler wrote it three years later, in 1782,
just a year before he died, and for some unknown reason he wrote it in French. The substitutions are
slightly different and the resulting algorithm is a bit more streamlined. Moreover, he takes less care to
get integer results. He is happy to get rational results, then multiply through by a common denominator
7
to make them integer. He gets yet again his examples 1 and 2 above, but this time he gives some new
examples, including
5. a = 159 b = 325 c = 314 x = 309.5 y = 188.5 z = 202
From this series of papers, we see that even near the end of his life, Euler went back over his
earlier results and tried to improve them. His blindness did not impair his amazing powers of
calculation or his ability to design ingenious substitutions. Moreover, while his students mostly worked
on applied problems, Euler seemed happy to work also on whimsical problems like this, just because
they were fun.
References:
[L] Loomis, Elisha S., The Pythagorean Proposition: Its Demonstrations Analyzed and Classified, And Bibliography of
Sources For Data of The Four Kinds of "Proofs", 2nd ed., Edwards Brothers, Ann Arbor, MI, 1940. (371 different
proofs of the Pythagorean theorem. First edition had 230.)
[M] Maor, Eli, The Pythagorean Theorem: A 4,000-Year History, Princeton University Press, 2007.
[E451] Solutio problematis de inveniendo triangulo, in quo rectae ex singulis angulis latera opposita bisecantes sint
rationales, Novi commentarii academiae scientiarum Petropolitanae 18 (1773) 1774, pp. 171-184 Reprinted in
Opera omnia I.3, pp, 282-296. Available at EulerArchive.org.
[E713] Euler, Leonhard, Investigatio trianguli in quo distantiae angulorm ab eius centro gravitates rationaliter exprimantur,
Nova acta academiae scientiarum Petropolitanae, 12 (1794) 1801 pp 68-69, 101-113. Reprinted in Opera omnia I.4
pp. 290-302. Available at EulerArchive.org.
[E732] Euler, Leonhard, Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita
bisecantes rationaliter exprimantur, Mémoires de l'académie des sciences de St.-Pétersbourg, 2 (1807/1808) 1810,
pp. 10-16. Reprinted in Opera omnia I.4, pp. 399-405. Available at EulerArchive.org.
[E754] Euler, Leonhard, Problème de géométrie résolu par l'analyse de Diophante, Mémoires de l'académie des sciences de
St-Pétersbourg 7 (1815/16), 1820, pp. 3-9. Opera omnia series I volume 5 pp. 28-34. . Available at
EulerArchive.org.
Ed Sandifer (SandiferE@wcsu.edu) is Professor of Mathematics at Western Connecticut State
University in Danbury, CT. He is an avid marathon runner, with 35 Boston Marathons on his shoes, and
he is Secretary of The Euler Society (www.EulerSociety.org). His first book, The Early Mathematics of
Leonhard Euler, was published by the MAA in December 2006, as part of the celebrations of Euler’s
tercentennial in 2007. The MAA published a collection of forty How Euler Did It columns in June
2007.
How Euler Did It is updated each month.
Copyright ©2008 Ed Sandifer