This may be proved independently of the preceding proposition by drawing the altitudes p and p'. Then
⧍ABC/⧍A'B'C' = cp/c'p'.
But c/c' = p/p',
by similar triangles.
∴ ⧍ABC/⧍A'B'C' = c2/c'2,
and so for other sides.
This proof is unnecessarily long, however, because of the introduction of the altitudes.
In this and several other propositions in Book IV occurs the expression "the square on a line." We have, in our departure from Euclid, treated a line either as a geometric figure or as a number (the length of the line), as was the more convenient. Of course if we are speaking of a line, the preferable expression is "square on the line," whereas if we speak of a number, we say "square of the number." In the case of a rectangle of two lines we have come to speak of the "product of the lines," meaning the product of their numerical values. We are therefore not as accurate in our phraseology as Euclid, and we do not pretend to be, for reasons already given. But when it comes to "square on a line" or "square of a line," the former is the one demanding no explanation or apology, and it is even better understood than the latter.
Theorem. The areas of two similar polygons are to each other as the squares on any two corresponding sides.
This is a proposition of great importance, and in due time the pupil sees that it applies to circles, with the necessary change of the word "sides" to "lines." It is well to ask a few questions like the following: If one square is twice as high as another, how do the areas compare? If the side of one equilateral triangle is three times as long as that of another, how do the perimeters compare? how do the areas compare? If the area of one square is twenty-five times the area of another square, the side of the first is how many times as long as the side of the second? If a photograph is enlarged so that a tree is four times as high as it was before, what is the ratio of corresponding dimensions? The area of the enlarged photograph is how many times as great as the area of the original?