MRF2017 wrote: Given a convex quadrilateral $ABCD$.Let $\ell_A,\ell_B,\ell_C,\ell_D$ be exterior angle bisectors of quadrilateral $ABCD$. Let $\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N$.Prove that if circumcircles of triangles $ABK$ and $CDM$ be externally tangent to each other then circumcircles of the triangles $BCL$ and $DAN$ are externally tangent to each other.(L.Emelyanov) Hint : Quote: if $\l'_A,\l'_B,\l'_C,\l'_D$ were the $interior$ angle bisectors of quadrilateral $ABCD$ $\l'_A \cap \l'_B=K',\l'_B \cap \l'_C=L',\l'_C \cap \l'_D=M',\l'_D \cap \l'_A=N'$ then $K'L'M'N'$ is inscribed in a circle