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 all lengths change by exactly the same multiplying factor. 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 a triangle is given by
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, if the line DE is drawn parallel to the base AC of triangle ABC, as follows
Then we claim that the sides AB and BC are divided proportionally as
where “m” is some positive real number. So that we can divide the two equations
To prove this proposition, we draw the lines AE and CD and lastly FE which is perpendicular to AB.
Then the areas of the specific triangles with a common height FE, can be written
Note that we could also draw a line from point “D” perpendicular to the line BC. Then by the same logic as above we can write
Also please note the triangles ADE and CDE 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 primary 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 and arrange their bases along the same line AE.
Then by extending the lines AB and DE to the point “F” we have created a parallelogram of BCDF which must therefore have equal length sides as
Using the result from Proposition 2, we note that the line BC is parallel to the line EF which is the base of the triangle AEF so that we can write
Or by simply rearranging
Again using Proposition 2, we note that the line CD is parallel to the line AF which can also be considered a base of triangle AEF. So we have
Or combining everything, we have
Or namely, that if we change the size of a triangle without changing any of its angles, then the lengths of all the sides are multiplied by exactly the same factor. And this is the fundamental assumption of trigonometry.