Clarification on a step of the proof of the 4 vertex theorem

Question

We are given a simple plane closed convex curve. Its curvature, being continuous on the entire curve, which is compact, achieves its maximum and minimum there, say at points $p$ and $q$. The author claims that the line which connects $p$ and $q$ divides the curve in two arcs, each in a definite side of that line, but I don't understand the reasoning.
He reasons by contradiction; he assumes it doesn't, and, as such, the line intersects the curve at some point $r$, distinct from $p$ and $q$. Here's the part I don't understand: he claims that, if that is the case, by convexity, somehow, the tangent to the intermediate point, say, $p$, must agree with the aforementioned line; and then he further states that again by convexity the line must be tangent to the curve at all points $p,q,r$. Why? To my understanding even if the tangent to $p$ agrees with this line, convexity is still ruined no?