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Transcript
THERE IS NO SSAAA CONGRUENCE THEOREM (5-CON TRIANGLES)
ANTONELLA PERUCCA
Consider two triangles such that two pairs of sides are equal in length, and all angles are
equal in measurement. The triangles must be similar because they have the same angles
but they need not to be congruent, as the following example shows.
λ
λ
λ3
λ2
2
λ
1
Take a triangle with sides
(1, λ, λ2 )
for some positive real number λ. Rescaling it by λ we obtain the sides
(λ, λ2 , λ3 )
thus all angles and two sides are the same. The two triangles are similar but they are not
congruent as soon as λ 6= 1. Up to switching the triangles, we may suppose λ > 1.
We have to choose λ carefully. Indeed, a triple (a, b, c) of positive real numbers such
that a 6 b 6 c consists of the sides of a triangle if and only if c < a + b holds. In our
situation (since we are assuming λ > 1) we must have λ2 < 1 + λ and hence
√
1+ 5
1<λ<
∼ 1.6 .
2
All problematic examples are of this form: the sides must be respectively
(r, rλ, rλ2 )
and
(rλ, rλ2 , rλ3 )
for some positive real numbers r and λ. We must require λ 6= 1 because we do not
want congruent triangles. It is crucial that two side ratios equal the similarity factor λ
because after rescaling we need to obtain two sides again.
Thus we must have two similar triangles that are not isosceles and such that the geometric mean of the greatest and the smallest side equals the third side (and the similarity
2
ANTONELLA PERUCCA
factor must be the square root of the ratio between the greatest and the smallest side).
• By setting r = 1000 and λ = 1.1, we obtain two acute triangles with integer sides:
1210
1100
1331
1210
1100
1000
• By setting r = 1000 and λ = 1.5, we obtain two obtuse triangles with integer sides:
3375
2250
2250
1500
1500
1000
• A triple (a, b, c) of positive real numbers such that a 6 b 6 c consists of the sides of
a right triangle if and only if a2 + b2 = c2 holds. Thus we can find a right triangle with
sides (1, λ, λ2 ) and hypothenuse λ2 because there is a positive real number λ satisfying
12 + λ2 = λ4 .
√
We first solve the quadratic equation 1 + x = x2 , which gives x = 1±2 5 . Setting the
positive solution equal to λ2 , we get
s
√
1+ 5
λ=
∼ 1.3 .
2
With this choice of λ and by setting r = 1 we obtain an example consisting of two right
triangles, namely the one displayed in the first figure.
Specifying three angles and two sides does not determine a triangle up to congruence,
but only up to similarity. There are problematic examples among acute/right/obtuse
triangles and among triangles with integer sides.
THERE IS NO SSAAA CONGRUENCE THEOREM (5-CON TRIANGLES)
3
.
5-Con triangles (published in Wikipedia.En)
Two triangles are said to be 5-Con or almost congruent if they are not congruent triangles but they are similar triangles and share two side lengths (of non-corresponding
sides). The 5-Con triangles are important examples for understanding the solution of
triangles. Indeed, knowing three angles and two sides is not enough to determine a
triangle up to congruence. A triangle is said to be 5-Con capable if there is another
triangle which is almost congruent to it.
The 5-Con triangles have been discussed by Pawley [1], and later by Jones and Peterson
[2]. They are briefly mentioned by Martin Gardner in his book Mathematical Circus.
Another reference is the following exercise [3]:
Explain how two triangles can have five parts (sides, angles) of one triangle congruent to five parts of the other triangle, but not be congruent
triangles.
A similar exercise dates back to 1955 [4], and there an earlier reference is mentioned.
It is however not possible to date the first occurrence of such standard exercises about
triangles.
8
12
18
12
18
27
F IGURE 1. The smallest 5-Con triangles with integral sides.
E XAMPLES
• There are infinitely many pairs of 5-Con triangles, even up to scaling.
• The smallest 5-Con triangles with integral sides have side lengths (8;12;18) and
(12;18;27). This is an example with obtuse triangles.
4
ANTONELLA PERUCCA
• Acute 5-Con triangles are for example (1000; 1100; 1210) and (1100; 1210; 1331).
2
2
3
• Right
q √5-Con triangles are for example (1; m; m ) and (m; m ; m ) with m =
1+ 5
. The right 5-Con triangles are exactly those obtained from scaling this
2
pair. There is no example of right 5-Con triangles with integral sides.
• There are no 5-Con triangles that are equilateral, or isosceles.
R ESULTS
(1) Consider 5-Con triangles with side lengths (a; b; c) and (ma; mb; mc) where m
is the scaling factor, which we may suppose to be greater than 1. We may also
suppose a ≤ b ≤ c. Then we must have b = ma and c = mb. The two triples of
side lengths are then of the form
a(1; m; m2 )
and
a(m; m2 ; m3 ) .
√
Conversely, for any a > 0 and 1 < m < 1+2 5 , such triples are the side lengths
for 5-Con triangles (Supposing w.l.o.g. a = 1, the greatest number in the first
triple is m2 and we only need to ensure m2 < 1+m; the second triple is obtained
from the first by scaling with m. So we have two triangles: They are clearly
similar and exactly two of the three side lengths coincide.).√ Some references
work with m−1 < 1 instead, which leads to the inequalities 5−1
< m−1 < 1.
2
(2) Any 5-Con capable triangle has different side lengths and the middle one is the
geometric mean of the other two. The ratio between the largest and the middle
side length is then equal to that between the middle and the smallest side length.
We can use both this ratio and its inverse for scaling and obtaining an almost
congruent triangle.
(3) To study the possible shapes of 5-Con triangles, we may restrict to studying the
triangles with side lengths
√
1+ 5
2
(1; m; m )
where
1<m<
.
2
The greatest angle is a strictly increasing continuous function of m and varies
from 60 to 180q
(the limit cases are excluded). The right triangle corresponds to
√
the value m = 1+2 5 . For convenience, scale the triangle to obtain (m−2 ; m−1 ; 1),
so that the largest side is fixed: The opposite vertex then moves along a curve
by varying m.
(4) Having two 5-Con triangles with integral sides√amounts (in the above notation)
to taking any rational number 1 < m < 1+2 5 and then choosing a > 0 in
such a way that am3 is an integer. The four involved integral side lenghts
(a; am; am2 ; am3 ) do not share any common factor (the 4-tuple is then called
primitive) if and only if they are of the form (x3 ; x2 y; xy 2 ; y 3 ) where x, y are
coprime positive integers.
THERE IS NO SSAAA CONGRUENCE THEOREM (5-CON TRIANGLES)
5
F IGURE 2. 5-Con triangles with the same greatest side.
F URTHER R EMARKS
Defining almost congruent triangles gives a binary relation on the set of triangles. This
relation is clearly not reflexive, and it is symmetric. It is not transitive: As a counterexample, consider the three triangles with side lengths (8; 12; 18), (12; 18; 27), and
(18; 27; 40.5).
There are infinite sequences of triangles such that any two subsequent terms are 5-Con
triangles. It is easy to construct such a sequence from any 5-Con capable triangle: To
get an ascending (respectively, descending) sequence, keep the two greatest (respectively, smallest) side lengths and simply choose a third greater (respectively, smaller)
side length to obtain a similar triangle. One may easily arrange the triangles in the
sequence in a neat way, for example in a spiral [1].
One generalization is considering 7-Con quadrilaterals, i.e. non congruent (and not
necessarily similar) quadrilaterals where four angles and three sides coincide or, more
generally, (2n-1)-Con n-gons [1].
R EFERENCES
[1] Pawley, Richard G. (1967). 5-Con triangles. The Mathematics Teacher. National Council of Teachers of Mathematics. 60 (5, May 1967): 438?443. Retrieved 6 March 2017.
[2] Jones, Robert T.; Peterson, Bruce B. (1974). Almost Congruent Triangles. Mathematics Magazine.
Mathematical Association of America. 47 (4, Sep. 1974): 180?189. Retrieved 6 March 2017.
[3] School Mathematics Study Group. (1960). Mathematics for high school–Geometry. Student’s text.
Geometry. 2. New Haven: Yale University Press. p. 382.
[4] Thebault, Victor; Pinzka, C. F. (1955). E1162. The American Mathematical Monthly. Mathematical
Association of America. 62, Dec. 1955 (10): 729?730. Retrieved 6 March 2017.
6
ANTONELLA PERUCCA
F IGURE 3. Two 7-Con quadrilaterals.