How was substitution chosen in ellipse equation derivation?

[symbolipoint]

I have been spending a few days reviewing parts of College Algebra from College Algebra, by Aufmann, Barker, & Nation. I am following the discussion of the shape, features, and equation for an ellipse, and I understand the derivation well, EXCEPT that I do not know how the subsitution of b² = a² - c², where a is half the length of the major axis, b is half the length of the minor axis, and c is the length from the origin to either focus. The standard form for the ellipse being:
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

I tried drawing my own pictures, graphs, to figure out how the substitution was chosen, but I made no progress, although I see that the substituion works by just trusting it. The page in the book for this is 311 and 312.

This is NOT homework. I earned my credit in Pre-Calculus in college about three decades ago.

(This is my third edit. Sometimes the TEX equation works, sometimes the TEX equation stays with all the tags showing. This is inconsistant.)

 

[symbolipoint]

Moderator who helped me with the typesetting in my question, thanks for that. I also tried studying the TEX typesetting in this message: https://www.physicsforums.com/showthread.php?p=3466447#post3466447

Still, anyone who can answer my question, please help with the substitution choice understanding.

 

[symbolipoint]

Finally some good progress in understanding, essentially the same information as in the book, in this online article, http://en.wikipedia.org/wiki/Proofs_involving_the_ellipse

The reason for the substitution makes more sense now. Distances from one focus to point on ellipse plus from other focus to same point on ellipse are always 2a; when x=0, y=b; then both distances from foci to (0, b) are equal. This makes right angle ..., b²+c²=a².