Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane.
Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$
In other words, is there a Riemann sum with two terms that hits the bull's-eye?
EDIT: Prompted by a down-vote, maybe I should give some motivation:
It's not difficult to convince oneself (but not completely trivial to prove (*)) that the barycenter $\frac{1}{b-a}\int_a^b f(t) \, dt$ is a convex combination of two values of $f$, say $f(x)$ and $f(y)$ with $x<y$. But this doesn't answer my question, because depending on the weights we may be unable to find the partition point $c$.
For curves in $\mathbb{R}^n$, I could ask if there is a Riemann sum with $n$ terms that hits the bull's-eye, but dimension $n=2$ already seems tricky enough.
(*) Footnote: A theorem of Korobkov [1], improving a theorem of McLeod [2], says that if $F: [a,b] \to \mathbb{R}^n$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then $\frac{F(b)-F(a)}{b-a}$ is a convex combination of $n$ values of the derivative $F'$ (which is not necessarily continuous).
References:
[1] Korobkov, M.V. -- A generalization of Lagrange's mean value theorem to the case of vector-valued mappings. Sibirsk. Mat. Zh. 42 (2001), no. 2, 349--353, ii; translation in Siberian Math. J. 42 (2001), no. 2, 297--300, doi: 10.1023/A:1004889013835.
[2] McLeod, R.M. -- Mean value theorems for vector valued functions. Proc. Edinburgh Math. Soc. (2) 14 1964/1965, 197--209, doi: 10.1017/S0013091500008786.
