The colored areas, are they of the same size?
Regards
The colored areas, are they of the same size?
Regards
Obviously not. The lower region is strictly taller than the upper region everywhere.
I'd look at whether their vertical cross-sections are of the same size. If you draw a vertical line intersecting both yellow regions, ask whether its intersection with one of those regions is longer than its intersection with the other one.
Rotating the two 180 degrees and then translating the smaller circle down and making the black horizontal lines the same, you get two integrals:
$\int_{0}^t (r_i-\sqrt{r_i^2-x^2}\ ) \ dx$
For $i=1,2$ and fixed $t$. This is not gonna be equal for two different $r_i$.
Basically, the two circles never "cross" - they touch at a tangent at the origin - so one of these functions is always smaller than the other, so the integrals will always be different except when $t=0$.
Geometric statement: If two distinct circles are both tangent to line $\mathcal l$ at point $P$, then they cannot have another common point. Proof: The centers of the two circles must be on the perpendicular to $\mathcal l$ at $P$. If there was another point $Q$ in common between the two circles, then the centers of both circles would also be on the perpendicular bisector of $PQ$. This would mean the circles share a center, and share a point, so they must be the same circle.
1) Take two squares $A$ and $B$ and put a circle as big as possible inside each of them. Denote the circle inside $A$ with $C$ and the circle inside $B$ with $D$.
2) Calculate the areas of $A$, $B$, $C$ and $D$. Calculate $x = \frac{A - C}{4}$ and $y = \frac{A - D}{4}$ to get the area between the corner of a square and the circle in it.
3) The yellow areas in your figures are proportional to $x$ and $y$
4) If $x = y$ for every $A$ and $B$ so are the yellow areas are also the same.
It's just a layman shouting here, so correct me if I am wrong.