If a triangle is contained in an algebraic variety, is its span contained too?

Question

Let $X\subset \mathbb{P}^{n}$ be an irreducible projective variety. Let us suppose that there exist three distinct points $x,y,z\in X$ such that $$ \langle x,y\rangle\subset X, $$ $$ \langle x,z\rangle\subset X, $$ $$ \langle y,z\rangle\subset X. $$ Is it true that the plane generated by $x,y,z$ is contained in $X$?

Answer 1

No. A cubic surface in $\mathbb P^3$ is a counterexample.