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In the common region of two circle, $\triangle ABC$ has been drawn with its maximum area such that the proportion of the maximum area of $\triangle ABC$ and the area of small circle is equal to $\frac{3\sqrt3}{4}:\pi$. How many parts of area of small circle is inscribed in the large circle?

It is very unclear a matter for me that how I should construct the equilateral $\triangle ABC$ in the common region. Moreover, I couldn't figure out even the radius of the large circle. Will it affect the equilateral triangle?

SOURCE: BANGLADESH MATH OLYMPIAD

Any kind of help will be appreciated. Thanks in advance.

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closed as unclear what you're asking by Aretino, stressed out, Cesareo, Kemono Chen, Anirban Niloy Feb 19 at 19:04

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ @jens You can check it out in GeoGebra. I used the app and found that moving outside the point $Q$, $\angle C$ results in both increasing and decreasing. Main fact is that there is a huge mistake in my diagram setup. $\endgroup$ – Anirban Niloy Feb 16 at 15:49
  • $\begingroup$ As I see it, if $Q$ was placed at $O$, the ratio of the common area would be $1$. As $Q$ is moved to the right along the line $OC$ and with the $Q$ circle still intersecting the same points $A$ and $B$, the ratio will decrease until the arc of the $Q$ circle between $A$ and $B$ approaches a straight line. $\endgroup$ – Jens Feb 16 at 16:40
  • $\begingroup$ I edited my diagram and brought a little change in my text and stated the point $D$ remaining still at same position. Keeping the point $D$ constant, I wanted to said about the increasing and decreasing of the radius of the large circle that obviously results in the same case for $OD$. And I told the ratio of $\triangle ABC$ and the area of small circle. And there is a little difference of the position of $A$ and $B$ by moving point $Q$. $\endgroup$ – Anirban Niloy Feb 16 at 17:09
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    $\begingroup$ Is it possible to have the actual text of the question? What you wrote is certainly not enough to give a meaningful answer. $\endgroup$ – Aretino Feb 16 at 17:18
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    $\begingroup$ Your question remains unclear. I also suggest you post the original text of the question. $\endgroup$ – Jens Feb 16 at 18:04