Flex point on an elliptic curve I have just started working through Pete Clark's elliptic curve notes, which are available here:
http://alpha.math.uga.edu/~pete/EllipticCurves.pdf
Early on, in section 2.1 on page 6, it is shown that the group law on $E(K)$ actually gives a group. Our distinguished point, $O$, is not assumed to be a flex point. Exercise 2.1 asks to show the following are equivalent:
i) $O$ is a flex point of the cubic curve $E$: i.e., the tangent line to $O$ intersects $O$ with multiplicity exactly 3.
ii) $O+O=O$.
iii) For all $A\in E(K)$, the inverse of $A$ lies on the line from $A$ to $O$.
Is part ii) correct? It was shown just before this exercise that $O$ is the identity for this group, and it isn't assumed to be a flex point. If $O$ is not a flex point, then to find $O+O$ we take the tangent at $O$, which by assumption intersects some other point $S\in E(K)$ with $S\neq O$. We then take the line from $S$ to $O$ and define $O+O$ to be the unique third point on this line. But this should just be $O$.
Is this correct?
 A: I agree with the OP and Bruno Joyal that the statement of this exercise is faulty.  As you say, condition (ii) had better hold for any rational point $O$ given that we've defined the group law in such a way to make $O$ the origin.  Unfortunately I could not remember what I had in mind when I wrote this, so I uploaded a new copy in which condition (ii) is simply removed.  The exercise still seems like a reasonable one: it shows why it is convenient to take the origin to be a rational flex point on the plane cubic...if you have one.  
Note that although the Fermat cubic $X^3+Y^3+Z^3 = 0$ has exactly three points over $\mathbb{Q}$, these points do not lie on a single line...nor could they, according to the exercise.
The idea of someone using these notes to learn about elliptic curves made me a little nervous.  I went back and added the following paragraph at the very beginning:

WARNING: These are the supplementary lecture notes for a first graduate course on elliptic curves (Math 8430) I taught at UGA in Fall 2012.  The word ``supplementary'' here is key: unlike most graduate courses I've taught in recent years, there was an official course text, namely [AEC].  Thus although the notes include what was discussed in the lectures, in their detailed coverage they tend to focus on slightly different material and/or a slightly different perspective than what is given in [AEC].  I want to emphasize that to read these notes without having the wonderful, classic text [AEC] in hand would be rather strange and is not recommended.

