Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE = A'$, $ CE\cap DA = B'$, $ DA\cap EB = C'$, $ EB\cap AC = D'$ and $ AC\cap BD = E'$. Suppose also that $ eABD'\cap eAC'E = A''$, $ eBCE'\cap eBD'A = B''$, $ eCDA'\cap eCE'B = C''$, $ eDEB'\cap eDA'C = D''$, $ eEAC'\cap eEB'D = E''$. Prove that $ AA'', BB'', CC'', DD'', EE''$ are concurrent. (Here $ l_1\cap l_2 = P$ means that $ P$ is the intersection of lines $ l_1$ and $ l_2$. Also $ eA_1A_2A_3\cap eB_1B_2B_3 = Q$ means that $ Q$ is the intersection of the circumcircles of $ \Delta A_1A_2A_3$ and $ \Delta B_1B_2B_3$.)