The following is a geometry problem that I came across with in the course of a research project.
Consider a ray starting at some initial point $t$. Place point $s_1$ at distance $r$ from $t$ on the ray and draw a circle centered at $s_1$ that passes through $t$. Likewise, centered at $t$, an arc with radius $r$ goes through $s_1$. Let $\mathcal{A_1}$ be the area enclosed between the intersecting arcs.
Next, arbitrarily place another point somewhere on the free end of the ray and call it $s_2$ such that $|s_1 - t| < |s_2 - t|$, where $|.|$ denotes the Euclidean distance. A circle with radius $r$ is centered at $s_2$ and another arc centered at $t$ goes through $s_2$. The area enclosed between these intersecting arcs we call $\mathcal{A}_2$. It is easy to show that $\mathcal{A}_1 < \mathcal{A}_2 < \lim_{|s_2 - t| \to \infty} \mathcal{A}_2 = \frac{1}{2} \pi r^2$.
Now, assume that we mark the segments of the ray within the enclosed areas in the middle and arcs centered at $t$ pass through the marks segmenting $\mathcal{A}_1$ and $\mathcal{A}_2$. We call these segmented areas $\mathcal{A}_{11}$ and $\mathcal{A}_{12}$ and $\mathcal{A}_{21}$ and $\mathcal{A}_{22}$ as depicted below (dashed lines are the arcs centered at $t$).
Question: How does $\mathcal{A}_{22}$ change as $s_2$ gets farther from $t$? (i.e., does it increase or decrease?) What can we say about $\mathcal{A}_{22}$ in comparison with $\mathcal{A}_{12}$?
Any idea or comment is much appreciated.
EDIT: The question has been edited in a way that makes the comments incomprehensible. Please see the edit history if you want to make sense of the comments.
EDIT: Here is the link to the same question at mathoverflow.net