See the figure Three lines through center of pair opposite red circle are concurrent (This is a statement of Dao's theorem on six circumcenter). The green circles are inverse of red circles in black circles. I am looking for a proof that: Three lines through center of pair opposite green circle are concurrent.

Let $L_1$, $L_2$, $L_3$, $L_4$, $L_5$, $L_6$ be six lines and let $P_{ij}= L_i \cap Lj$, such that $P_{12},P_{23}, P_{34}, P_{45}, P_{56}, P_{61}$ lie on a circle $(\Omega)$. Let $(O_{ijk})$ is inverse of circle $(P_{ij}P_{jk}P_{ik})$ in $(\Omega)$. We taking subscripts modulo 6. Then three lines $O_{126}O_{453}$, $O_{231}O_{564}$ and $O_{342}O_{651}$ are concurrent