Proportionality of Similar Triangles
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
If we draw a line through a triangle which is parallel to one of the sides, then it is claimed that line divides the other two sides in equal proportions. For example, we can imagine a generalized triangle ABC of arbitrary size. Then we draw the line DE parallel to the base AC as follows
The basic claim is that the sides AB and BC are divided proportionally as
where “m” is some positive real number. If we divide the two equations, we get the fundamental assertion
To prove this proposition, we draw the line AE and then EF which is perpendicular to AB.
The areas of the triangles with a common height EF, can be written
We can also draw the line CD and then DG which is perpendicular EB.
Then by the same logic as above we can write
Also please note the triangles DEC and DEA have the same area because they have the same base DE and the same height, “h”.
Thus we can write
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. And this is the fundamental assumption of trigonometry.