Proportionality of Similar Triangles

INTRODUCTION

The fundamental assumption of trigonometry is that if a triangle, or indeed any polygon, changes its size but not its shape, then the lengths of all the edges are multiplied by exactly the same factor.  This means, for instance, that if one edge doubles in length, then all edges double in length.  And so the ratio of any two lengths remains exactly the same as the figure shrinks or expands.

This is important enough for a rigorous proof rather than being left as simply an appeal to fuzzy minded reasoning.  Towards that end, we copy two proofs from Euclid’s Elements Book VI namely Propositions 2 and 4.

Euclid was a Greek mathematician who lived in Alexandria, Egypt about 300 BC.  His work consists of 13 Books and provides the first logical development of geometry by reasoning from a simple set of axioms.    As such it is one of the more influential mathematical works ever written.

Please recall the area “A” of any triangle is given by the formula

EUCLID’S CLAIM OF BOOK VI PROPOSITION 2

Consider a general triangle ACE of arbitrary size as shown below.   The line extended through BD is parallel to the base of the triangle AE.  Also the perpendicular distance from the point B to the line CE is h1 and from point D to AC is h2.

We can then write the areas “A” of various smaller included triangles as follows

The since AABD = AEBD, we can write the ratios with respect to the top triangle ABCD

Or finally we note the line BD cuts the two sides of the triangle ACE in equal proportion as

EUCLID’S CLAIM OF BOOK VI PROPOSITION 4

We next make the central claim of this exercise, that if we have two “similar” triangles of different sizes, then the lengths of their sides are all in the same proportion.   By “similar” triangles we mean they have the same shape.   The minimum requirement for being similar is that at least two angles in each triangle are equal to those in the other triangle.   Since all three angles in every triangle sum to 180°, if two angles are equal, then all three angles must be equal.

To prove this we first draw two similar triangles ABC and CDE which have different sizes but the same shape and thus the same angles and arrange their bases along the same straight line AE.

Then by extending the lines AB and ED to the point “F” we have created a parallelogram of BCDF with two parallel and equal sides.

Using the result from Proposition 2, we note that the line BC is parallel to the line EF so that we can write

Again using Proposition 2, we note that the line CD is parallel to the line AF so we have

We also note that the parallelogram BCDF has equal length sides as

Or combining everything, we have

Or namely, that if we change the size of a triangle without changing any of its angles, as for instance the triangles ABC and CDE, then the ratios of the lengths of all the sides are exactly the same.   That is any expansion or contraction of the basic shape is linear.  And this is the fundamental assumption of trigonometry.