  Let $Y \subset \P^r$ be a normal nondegenerate m-dimensional subvariety and
let $\sigma(Y)$ denote the maximum dimension of a subvariety $Z \subset
Y_{smooth}$ such that $Z$ contains a generic point of some divisor on $Y$ and
the tangent planes $T_y Y$ for all $y \in Z$ are contained in a fixed
hyperplane. In this article we study the double locus $D \subset $Y$ of its
generic projection to $\P^{r-1}$, proving that if the secant variety of $Y$ is
the whole space and $\sigma(Y) < 2m - r + 1$, then $D$ is irreducible. Applying
Zak's Tangency theorem we deduce the irreducibility of $D$ when $m > 2(r-1)/3$.
The latter implies a version of Zak's Linear Normality theorem.
