Geometric proof of Convexity iff all chords are contained

I have this problem:

Let CC a simple and closed plain curve parametrized by (I,α)(I,\alpha).

Suppose that CC is convex. Demostrate that if the chord ¯α(t)α(t′)\overline{\alpha(t)\alpha(t’)} intersects CC in other point α(t″)\alpha(t”), then ¯α(t)α(t′)\overline{\alpha(t)\alpha(t’)} is contained in CC.
Let AA be the closure of the bounded connected component of \mathbb{R}^2 \setminus C \mathbb{R}^2 \setminus C . Get from the first statement that CC is convex iff all chords \overline{\alpha(t)\alpha(t’)}\overline{\alpha(t)\alpha(t’)} are contained in AA.

I know that this can be proved by using neighborhoods and the convexity of the set, but I want to know if there exists a proof of this using only geometric arguments.

Thanks.

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