# Segmented area between circles

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 $A_1$ 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 $A_2$. It is not hard to show that $A_1 < A_2 < \lim_{|s_2 - t| \to \infty} A_2 = \frac{1}{2} \pi r^2$.

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 $A_1$ and $A_2$. We call these segmented areas $A_{11}$ and $A_{12}$ and $A_{21}$ and $A_{22}$ as depicted here (dashed lines are the arcs centered at $t$).

Question: How does $A_{22}$ change as $s_2$ gets farther from $t$? (i.e., does it increase or decrease?) What can we say about $A_{22}$ in comparison with $A_{12}$?

Any idea or comment is much appreciated.

P.S. - Here is the link to the same question at math.stackexchange.com

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Ali, please edit in a link to the identical question asked at math.stackexchange.com. Then go back there and edit in a link to this question. –  Gerry Myerson Jun 21 '12 at 5:29
Ali, why don't you just compute the area $A_{22}$ in terms of the distance of centers, and check that it is increasing? –  Pietro Majer Jun 21 '12 at 5:57
@Gerry Myerson: Sorry, do you mean I cross reference both questions in two sites? –  Ali Jun 21 '12 at 5:58
@Pietro Majer: It does not look trivial to me how that helps simplify the question. Can you please explain further? –  Ali Jun 21 '12 at 6:04
Sure: compute the area $A_{2,2}$ as a function of $r$ and of the distance between the centers $x:=|t-s_2|$. Then, check that this function is increasing wrto $x$. –  Pietro Majer Jun 21 '12 at 6:18