Are there distinct orthogonal lines to simple curve intersecting at a point *inside* the curve?

Question

The orthogonal lines to a semicircle meet in the center of the diameter. I'm trying to generalize a similar statement for any curve between two points.

Consider two points $a$ and $b$ along a line, and an arbitrary simple curve $C$ between them, as shown below. Are there always two perpendiculars to $C$ ($R$ and $S$ in the picture), such that their intersection point $p$ lies inside the domain bounded by $C\cup\overline{ab}$ ?

In the limiting case of the semicircle, $p$ lies along $\overline{ab}$ and all the lines pass through $p$, but it's hard to generalize solely from the knowledge of $C$ being a non-self-intersecting curve.

(Perpendicular to $C$ means perpendicular to the tangent to $C$ at the point).

Answer 1

Any circle arc from $a$ to $b$ of smaller angle than the semicircle is a counterexample.

This is because the perpendiculars to tangents will be the circle center, and the center will not be outside the region between the arc and the chord $\overline{ab},$ for arcs of angle $\pi.$

Answer 2

Consider any curve with a section parallel to the line $\overline{ab}$. Both orthogonal lines on that section will be parallel and so they won't intersect at all.