Let $X$ be a (smooth, irreducible) curve of degree $d$ in $\mathbb{P}^2_k$ where $k$ is an algebraically closed field of characteristic $0.$ We say that a line of $\mathbb{P}^2_k$ is a multiple tangent of $X$ if it is tangent to $X$ at more than one point. If $L$ is a multiple tangent of $X,$ tangent to $X$ at the points $P_1, \ldots, P_r$ and if none of the $P_i$ is an inflection point, show that the corresponding point of the dual curve $X^*$ is an ordinary $r$-fold point, meaning that it is a point of multiplicity $r$ with distinct tangent directions.

This is a question from Hartshorne, and while I believe I know how to prove it (with different methods), I don't think is the kind of solution the exercise warrants. Let me briefly outline the methods I know of how to prove this:

- Assume that we know that the Gauss map $X \rightarrow X^*$ is birational. Then it follows that $X$ is the normalization of $X^*$ and we can identify the Gauss map with successive blow-ups at singularities of $X^*.$ Then one checks that essentially by definition that the point $L$ is an ordinary $r$-fold point. To know that the Gauss map is birational, one can argue by the biduality theorem for plane curves in characteristic $0.$
- Use the Lefschetz principle to reduce to when $X$ is a curve over $\mathbb{C}$ and argue using the analytic topology / by geometry.
- Pass to completions and try to define the Gauss map locally.

All of these methods seems to me to be somewhat against the spirit of the material that Hartshorne has introduced so far in the book. The first of the above items uses the biduality theorem, which, while not extremely hard to prove, still is quite a lot for this The second is not algebraic at all the last seems somewhat of a stretch.

I would be very grateful for a purely algebraic solution of this exercise. Let me explain what I mean by a purely algebraic solution. In some sense, it should only use elementary machinery from scheme theory and not go to the analytic category. If completions are neccessary, they can be used but I would prefer something not using that. Further, I would (if possible) want it to be a proof that would hold true in characteristic $p>0$ as well.

This question has previously been posted to math.stackexchange, but the answers I received was not what I was looking for so I thought I would post it here. I apologize if this is not the right forum for this question, but I am really curious.