smaller ore equal circle than circumcircle wanted, <RMN= <RNM= <RPC= <RQC= 30^o

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smaller ore equal circle than circumcircle wanted, <RMN= <RNM= <RPC= <RQC= 30^o

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Source: 1992 IberoAmerican Shortlist p8 https://artofproblemsolving.com/community/c2468825_iberoamerican_shortlist__geometry

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#1 h Oct 6, 2021, 2:23 PM

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Let $A$ and $B$ be two points outside the circle $K$ . If we draw the segment $\overline{AB}$ , it intersects the circle at points $M$ and $N$ ( $M \in \overline{AN}$ ). We take a point $C$ inside the circle and draw the segments $\overline{AC}$ and $\overline{BC}$ , so that $\overline{AC}$ intersects the circle at $Q$ and $\overline{BC}$ at $P$ . If $M$ and $Q$ are equidistant from $A$ and $N$ , and $P$ from $B$ and there is a point $R$ interior to triangle $ABC$ such that $\angle RMN = \angle RNM = \angle RPC = \angle RQC = 30^o.$ Prove that the area of said circle is less than or equal to the area of the circle circumscribed to triangle $ABC$ .

This post has been edited 3 times. Last edited by parmenides51, Dec 15, 2022, 6:52 PM

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