Proportionality in Similar Triangles: A Cross-Cultural Comparison - Introduction

Editor’s note: This article was published in 2007.

A fundamental result of geometry, used often in secondary and collegiate mathematics, is the equality of ratios of corresponding sides in similar triangles. This concept is expected knowledge of students in physics, engineering, and the sciences, since its simple statement is rather useful in finding unknown lengths of elementary figures in plane geometry. Why does this geometric result hold? What do we teach our students about this relation, and, more importantly, how is this result presented to secondary education majors, who will be teaching future college students? Is the result justified by an appeal to reason, or are similar triangles defined as those triangles for which corresponding sides have equal ratios? Both of these approaches are found in today's curriculum, although neither is particularly revealing.

This article offers curricular materials for the proof of similarity theorems, based on an ancient Chinese principle of area known as the "in-out" or "inclusion-exclusion" principle. When applied to a rectangle, the principle identifies certain (non-congruent) sub-rectangles of equal area that remain after the exclusion of congruent triangles. The principle is easily applied when the excluded triangles are right triangles, and to account for all possible pairs of corresponding sides in a right triangle, further application of the gou-gu (Pythagorean) theorem is used. Another engaging use of the inclusion-exclusion principle is the proof of the gou-gu theorem itself, borrowing an idea from the text Zhou bi suan jing (Mathematical Classic of the Zhou Gnomon) [5], compiled between 100 BCE and 100 CE. Strictly speaking gou refers to base or shadow and gu refers to height or gnomon, although there apparently was no word per se for the concept of a triangle in ancient China [5, p. 215]. See Mathematics in China <http://aleph0.clarku.edu/~djoyce/mathhist/china.html> at David Joyce's Mathematics History website <http://aleph0.clarku.edu/~djoyce/mathhist/> for further information about the history of Chinese mathematics.

Since teaching similarity results from ancient Chinese principles of reasoning may be novel to many instructors, the article begins with Euclid's classical approach to the subject. Although ideas of reasoning and a concept of rigor differ between the Chinese and Greek schools of mathematical thought, similarity results for both cultures rest ultimately on two-dimensional area arguments. A latent mathematical axiom behind any similarity result is the parallel postulate, a subtle axiom with implications impinging on many constructions and theorems in geometry. The existence of a rectangle itself, not to mention a formula for its area, is logically equivalent to the Euclidean parallel postulate [2]. Likewise the expression of the area of a triangle as one-half the base times the height is a Euclidean formula, since it relies on the area result of an enveloping rectangle or parallelogram. Euclid's proof of similarity results relies on proposition 38 from book I of The Elements [1]:

I.38. Triangles which are on equal bases and in the same parallels are equal to one another.

Given triangle ABC with side AB identified as the base, there is only one line through C parallel to AB, thus determining the height of the triangle. Proposition I.38 asserts that two triangles with equal bases and equal heights will in fact have equal area. Another result equivalent to the Euclidean parallel postulate is the theorem stating that any triangle has angle sum 180^° [2], which is in turn equivalent to a rectangle having angle sum 360^°. The web resource Non-Euclidean Geometry <http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html> at the MacTutor History of Mathematics Archive <http://www-history.mcs.st-and.ac.uk> offers further history of the parallel postulate.

While Euclid follows a step-by-step model of deductive reasoning with every statement justified by a previous proposition, definition, or postulate, the Chinese method is a bit more intuitive, particularly when identifying what today would be called congruent triangles. The ease by which similarity results are then proven (as a modern exercise) is appealing. Moreover, the argument founded on Chinese principles does not require a comparison of possibly two incommensurable lengths for the bases in similar triangles, as Euclid must consider. Two lengths L₁, L₂ are commensurable if a whole-number multiple of L₁ can be constructed on a whole-number multiple of L₂, or in modern language [3, p. 30], if there are positive integers n₁, n₂ with n₁ L₁ = n₂ L₂, or equivalently, [(L₁)/(L₂)] is a rational number. See the web resource Greek Mathematics <http://www-history.mcs.st-and.ac.uk/Indexes/Greeks.html> at the MacTutor History of Mathematics Archive <http://www-history.mcs.st-and.ac.uk> for detailed information about the history of Greek mathematics.

The curricular presented materials in Section 4 are ideal for a course in geometry, taught either in college or high-school, or for a course that draws prospective teachers of secondary mathematics. For use in the classroom, the instructor should present the results of Section 3, although Section 2 may be omitted from class discussion, depending on course direction and time constraints. When assigning the material in class, the instructor may delete or rearrange certain parts of the teaching module to fit the course.