Let six lines $L_1, L_2, L_3, L_4, L_5, L_6$ through point $O$ and $L'_1, L'_2, L'_3, L'_4, L'_5, L'_6$ through point $O'$, Let $P_{ij}=L_i \cap L'_j$. Such that $P_{ii}$ lie on a line for $i=1,2,3,4,5,6$.
1-Then show that: $P_{12}P_{21}, P_{34}P_{43}, P_{56}P_{65}$ are concurrent.
2-Let the point of concurrence is $P$ then show that $P, O, O'$ are collinear.
3-And $P_{12}, P_{21}, P_{34}, P_{43}, P_{56}, P_{65}$ lie on a conic.
I think the natural realm of this question is projective geometry, rather than Euclidean geometry as your tag currently indicates. So I'll be using a number of projective concepts in my answer, and probably edit the tag of the question.
Observe that the following is the standard construction of a harmonic range:
The position of $P$ in this construction only depends on $O,O',Q$ but not on $P_{11},P_{22},P_{21},P_{12}$. In your setup, you have essentially three instances of this construction, which explains why all three lines coincide with that point $P$ on the line $OO'$, proving the first two claims.
The third claim could be proven using coordinates. The whole setup is invariant under projective transformations, so without loss of generality, one can establish the following homogeneous coordinate system:
$$ O=\begin{pmatrix}0\\1\\0\end{pmatrix}\qquad P_{11}=\begin{pmatrix}a\\0\\1\end{pmatrix}\quad P_{22}=\begin{pmatrix}b\\0\\1\end{pmatrix}\quad P_{33}=\begin{pmatrix}c\\0\\1\end{pmatrix}\\ O'=\begin{pmatrix}0\\1\\1\end{pmatrix}\qquad P_{44}=\begin{pmatrix}d\\0\\1\end{pmatrix}\quad P_{55}=\begin{pmatrix}e\\0\\1\end{pmatrix}\quad P_{66}=\begin{pmatrix}f\\0\\1\end{pmatrix} $$
Then you have
\begin{align*} P_{12}&=\begin{pmatrix}ab\\b-a\\b\end{pmatrix}& P_{34}&=\begin{pmatrix}cd\\d-c\\d\end{pmatrix}& P_{56}&=\begin{pmatrix}ef\\f-e\\f\end{pmatrix}\\ P_{21}&=\begin{pmatrix}ab\\a-b\\a\end{pmatrix}& P_{43}&=\begin{pmatrix}cd\\c-d\\c\end{pmatrix}& P_{65}&=\begin{pmatrix}ef\\e-f\\e\end{pmatrix} \end{align*}
To show that these lie on a conic, one can check the following condition, using brackets to denote determinants:
$$ [P_{12}P_{34}P_{56}][P_{21}P_{43}P_{56}][P_{12}P_{43}P_{65}][P_{21}P_{34}P_{65}] \\ - [P_{12}P_{34}P_{65}][P_{21}P_{43}P_{65}][P_{12}P_{43}P_{56}][P_{21}P_{34}P_{56}] = 0 $$
Verification now becomes a simple exercise in multivariate polynomial expansion, although I'm too lazy to do this manually, so I had this verified by Sage.