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IstekOlympiadTeam wrote: Circle with center $O$ and radius $R$ intersects $AC$ at $K$. I realize this might be because you translated it, but what does this sentence mean?

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IstekOlympiadTeam wrote: Circle with center $O$ and radius $R$ intersects $AC$ at $K$.A. This circle is the circumcircle of $ABC$ and cannot intersects (or cuts) $AC$ in a different point $K$ from $A$ and $C$.

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Okay. I have made an investigation and come up with this correct configuration of what the problem could have been, and to make it look nice I changed it to an obtuse triangle. Here it is: An obtuse triangle $ABC$ with angle $B > 90$ degrees and $AC > BC > AB$ has the circumcircle $(O)$. $BD$ and $CE$ are diameters of $(O)$. Circle with center $A$ and radius $AD$ intersects ray $AB$ at $L$, and circle with center $A$ and radius $AE$ intersects ray $AC$ at $K$. Prove that lines $DL$ and $EK$ intersect on $(O)$.

NgoNgang wrote: Okay. I have made an investigation and come up with this correct configuration of what the problem could have been, and to make it look nice I changed it to an obtuse triangle. Here it is: An obtuse triangle $ABC$ with angle $B > 90$ degrees and $AC > BC > AB$ has the circumcircle $(O)$. $BD$ and $CE$ are diameters of $(O)$. Circle with center $A$ and radius $AD$ intersects ray $AB$ at $L$, and circle with center $A$ and radius $AE$ intersects ray $AC$ at $K$. Prove that lines $DL$ and $EK$ intersect on $(O)$. Nice work, but ... $D,E$ belong to $(O)$, so that intersection cannot belong to $(O)$! Best regards, sunken rock

Yes, it does because $K$ is inside $(O)$ and $L$ is outside $(O)$.