Prove $AX$, $BY$, $CZ$ concur

Question

$\Delta ABC$. Let $P,Q$ be two points lies in the interior of the triangle. Let $AP,BP,CP$ intersect $BC,CA,AB$ at $X_P,Y_P,Z_P$. Point $X_Q,Y_Q,Z_Q$ are defined similarly. Let $T_P$ be a point lies on $PQ$. Let $(C)$ be the conic pass through five points $Y_P,Z_P, X_Q, Y_Q, Z_Q$. Let $A',B',C'$ be the second intersection of $X_PT,Y_PY,Z_PT$ with $(C)$. Prove $AA',BB',CC'$ concur.

• What I have done:
  1. I proved the six points lies in conic by Carnot theorem
  2. Using Pascal theorem, I proved $X_QA',Y_QB',Z_QC'$ also concur at a point in $PQ$ (I called the concur point $T_Q$).
  3. I'm wondering do we have: $\dfrac{\sin{A'AB}}{\sin{A'AC}}=\dfrac{A'Z_Q}{A'Y_Q}.\dfrac{A'Z_P}{A'Y_P}$ (I) ?. We do have (I) when $(C)$ is a circle (proved) but when (C) just a conic, does (I) still true ?