By leveraging an inherently geometric argument, we establish new properties of convex functions \( \Gamma \) defined on a convex domain \( \Omega \). Suppose the graph of (convex) \( \Gamma \) contains a line segment \( [Y^1,Y^2] \), where \( Y^j=(y^j,\Gamma(y^j)) \), with \( y_j\in\Omega \), and that \( Y^*=(y^*,\Gamma(y^*)) \) lies on this segment. Given a second-order polynomial \( P \), whose graph touches (locally) the graph of \( \Gamma \) from below at \( Y^* \), then a horizontal/vertical translation of \( P/3 \) touches the convex graph of \( \Gamma \) at least at one of the points \( Y^j \). This, in light of the viscosity approach in PDEs, has interesting consequences for the regularity of convex envelopes of supersolutions to a large class of partial differential equations. These include degenerate fully nonlinear models and quasi-linear problems that have not been treated in the literature. Our methods are versatile, and we expect them to find applications in a broader class of models.