LEE Chun-yue Euclidean Definitions
Mathematics textbooks used in Hong Kong junior secondary schools often define similarity either as
1. Two figures are similar if they have the same shape and different sizes; or
2. Two figures are similar if they have the same shape.
Then theorems are established only for similar triangles – theorems that include
1. Two triangles are similar if and only if their corresponding angles are equal;
2. Two triangles are similar if and only if their corresponding sides are proportional.
The theorems are also stated for comparing volumes and surface areas of similar solids.
On the other hand, Euclid and authors of old geometry books(I) defined right away similar rectilinear figures by having the two criteria of
(a) all corresponding angles equal; and
(b) all corresponding sides proportional. (1) Mayne (1961) in fact defined same shape in terms of similar figures.
Similar figures are said to have the same shape. (p. 332) After his work on similar rectilinear figures, in book XI of his work, Euclid went on to define similar solids.
Two solids are similar if they are formed by the same number of similar planes(II).
Euclid’s definition of similarity has been well recognised, and it is common for textbook to introduce the definition by way of same shapes. But could similarity of rectilinear figures be defined in some other ways, without the use of the concept shape, and without the presence of the two criteria (a) and (b)?
To me, these questions interest me very much. After searching for the answers for quite a while, I discover that similarity of rectilinear figures can actually be defined by means of the concept of transformation, a topic newly introduced in the Hong Kong Secondary Mathematics curriculum in 1999(III). The following shows my argument.
A New Definition for Similarity
I would like to define similar rectilinear figures in the following way.
Two figures are similar if and only if one of the figures coincides with the other figure within finite number of times of reflection, rotation, translation and/or dilation transformation.
I assume reflection, rotation and translation to have their usual meanings, but dilation is defined here as either a magnification or a contraction(IV). Mathematically, a point A is transformed by dilation at a point O with a scaling factor k (k being a non-zero real number) to the image A on the line OA if only if OA : OA = k : 1.
Next, I want to prove that the similar figures under such a definition will have the properties that (a) the corresponding angles of similar figures are equal and (b) the sides about the equal angles are proportional.
I will not discuss reflection, rotation and translation because readers can easily verify that these transformations are rigid motions in which all lengths and angles are preserved. Instead, I will simply focus on dilation and prove the following proposition.
Proposition
In figure 1, AB is any line segment and O is a point of projection.
If AB is transformed by dilation at O with scaling factor k (k can be any real number except 0 or 1) to the image AB (i.e., for any point U on AB and U the image of U under the dilation, OU:OU
= k : 1. In particular, OA:OA = OB:OB = k : 1), then 1. AB // AB, and
2. k AB = AB
One should note that in usual practice, the results can easily be obtained by properties of similar triangles stated in (1).
However, it will fall into “circular reasoning”, since the properties of similar figures have not been derived from the definition yet. Hence, to avoid logical loopholes, I need to prove the proposition without using any knowledge of similar triangles. The next section is the proof.
O
A
B
A
B
Figure 1
Proof
1 We first prove that AB // AB.
Assume the contrary, AB and AB are not parallel but they intersect at a point X. See Figure 2.
Figure 2
Join AB and OX. For convenience, we denote the area of each triangle in the figure by a small letter. For example, the area of
OAB is denoted by p, the area of OBX by q etc. See figure 3.
X O
A
B A
B
Claim. If
and
, then
(1)
The proof is left to the reader as an exercise.
Now, using the theorem that areas of triangles of equal heights are proportional to their base lengths, and remembering that
1 : k ' OB : OB ' OA :
OA , we obtain
1
k q p
k r s t
（OA : OA = k : 1 k） (2)
1
k q
k r
（OB : OB = k : 1 k） (3) From (2) and (3) and comparing the result (1), replacing , ,
, , respectively by k k
1 , q , r , p , and s + t , we get X O
A
B A
B
Figure 3 p
q r
s p t k
1
k 1
t s
p k k
1 (4)
Also,
t p k k
1 （OA : OA’ = k : 1 k） (5) Applying result (1) again to (4) and (5), we get
s k
k 0
1
which cannot be true because k is non-zero.
Hence, AB // AB.
2 We then go on to prove k AB = AB , or AB : AB = k : 1, by applying the result AB // A’B’ which we have just shown. Let C be a point on A’B’ such that AC // BB’. Join AB’. See figure 4.
O
A
B
A
B
C
Figure 4 p
s k
1
k 1
q
q H
1
Clearly, ABBC is a parallelogram, AB = CB.
Suppose that AB : AB = CB : AB = H : 1.
It is required to show that H = k.
As before, we denote the area of each triangle by small letters, as shown in figure 4.
s q
q p k k
1 （OA : OA = k : 1） (6) q
p k k
1 （OB : OB = k : 1） (7) Applying result (1) to (6) and (7), we get
s q k k
1 (8)
But
s q H H
1 （CB : AB = H : 1） (9) Hence, H = k.
Hence, AB : AB = OA : OA = OB : OB = k : 1.
Thus, the proof is complete.
With this proposition, one can easily conclude that the dilation transformation is angle-preserving (note the parallel property which has been proved). Moreover, given a figure, if it is transformed by dilation all corresponding lengths are proportional. As a consequence, the following statement holds.
If a figure can be transformed by a finite number of times of reflection, rotation, translation and dilation to coincide with another figure, then the corresponding angles of these two figures are equal and all the corresponding lengths of the figures are proportional.
Summary
We have seen different ways in defining similarity of rectilinear figures. Table A is a summary of the three definitions discussed.
Table A
The Three Definitions of Similarity of Rectilinear Figures Textbooks Euclid My point of
view Two
Rectilinear Figures are Similar
Having the same shape
(a) All
corresponding angles equal and (b) All corresponding sides
proportional
Coincide after a finite number of reflection, rotation, translation and dilation Similarity of rectilinear figures can indeed be defined by means of transformation. The next question that arouses my interests is therefore: Could similarity of figures of a higher dimension be
defined by means of transformation? This is another good question that is worth studying.
Notes
(I)
Such a definition can be found in a book The Thirteen Books of Euclid's Elements (2nd ed., Vols. 1-13), New York: Dover, written in 1925 by Heath, Sir Thomas Little. The book was in fact translated from the text of Heiberg – published in 1908.
The definition can also be found in p.332 of The Essentials of School Geometry, written by Mayne in 1961, and published in London by Macmillan.
Such a definition can also be found in page 142 of the book 藍紀 正、朱恩寬（1992）。《歐幾里得幾何原本》。台北：九章。The book was translated from Heath’s book The Thirteen Books of Euclid's Elements (2nd ed., Vols. 1-13).
(II)
Such a definition can be found in page 463 of the book 藍紀正、
朱恩寬（1992）。《歐幾里得幾何原本》。台北：九章。The book was translated from Heath’s book The Thirteen Books of Euclid's Elements (2nd ed., Vols. 1-13).
(III)
Hong Kong Curriculum Development Council. (1999).
Syllabuses for Secondary Schools Mathematics Secondary 1-5.
Hong Kong: Printing Department.
(IV)
The Syllabuses for Secondary Schools Mathematics Secondary 1-5 used dilation/contraction (Hong Kong Curriculum Development Council, 1999, p. 21), but I prefer to use a single word dilation to stand for both magnification and contraction.