Abstract: In the figure belows: Three lines through center of pair opposite red circle are concurrent. This is a statement of Dao's theorem on six circumcenter, a new theorem in plane geometry which was discovered by OP in 2013. Also on this configuration, today 05/25/2021, I found that: Let the green circles are inverse of red circles in black circle $(\Omega)$. Then three lines through center of pair opposite green circle are concurrent.

I am looking for a proof that:

Problem: 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.