http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html
I have understood the proof in general. It is only a small detail which i'm not sure. Maybe it's because english isn't my first language. Anyway, the part of the proof which says : "Then, since BC is nearer to the center and FG more remote, EK is greater than EH." I have no problem understanding what is said here. This is supported by this definition : "And that straight line is said to be at a greater distance on which the greater perpendicular falls." (Definition 5 of book 3) Now, this is where i'm unsure. From what I understand of it, it says that if I have a perpendicular that is bigger than the other, than my straight line is said to be at a greater distance. (This is how I understand it) Now, in the proof, we do the inverse. We know that one line is at a greater distance than the other and we conclude with the definition that one perpendicular is bigger than the other. How is this correct ? Unless the definition implies that the reverse is also ok, then this works. But if the definition implies only one direction, (The one which is defined) then how is the proof valid ?
By the way, you don't need to read all of the proof. Only the things at beginning are needed.
Here's some text just in case the link doesn't work anymore :
"Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote. Let ABCD be a circle, AD its diameter, and E its center. Let BC be nearer to the center AD, and FG more remote. I say that AD is greatest and BC greater than FG. I.12 Draw EH and EK from the center E perpendicular to BC and FG. III.Def.5 Then, since BC is nearer to the center and FG more remote, EK is greater than EH. I.3 I.11 Make EL equal to EH. Draw LM through L at right angles to EK, and carry it through to N. Join ME, EN, FE, and EG. III.14 Then, since EH equals EL, BC also equals MN. Again, since AE equals EM, and ED equals EN, AD equals the sum of ME and EN. I.20 But the sum of ME and EN is greater than MN, and MN equals BC, therefore AD is greater than BC. I.24 And, since the two sides ME and EN equal the two sides FE and EG, and the angle MEN greater than the angle FEG, therefore the base MN is greater than the base FG. But MN was proved equal to BC. Therefore the diameter AD is greatest and BC greater than FG. Therefore of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote. Q.E.D."
This is just a matter of how definitions are formulated. A definitionis typically formulated with an if, but as the condition is taken as defining property, this if can actually be read as if and only if. Put differently, the only way for you to be sure that "$\ell_1$ is at greater distance than $\ell_2$" is to apply the definition of "is at greater distance", that is to rewrite this as "the perpendicular to $\ell_1$ is grater than the perpendicular to $\ell_2$"