Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle.
Prove that
\[AX+AY+BC>AB+AC\]
Say $AB$ is tangent to the incircle at $AE$. Then it suffices to prove $AX + AY > 2AE$. But $AX \times AY = AE^2$, so this follows by AM-GM (clearly $AX < AE$ so equality does not hold.)