I gave a generalization of the [Tucker circle theorem](http://mathworld.wolfram.com/TuckerCircles.html) and the [Thomsen theorem](https://en.wikipedia.org/wiki/Thomsen%27s_theorem) at [here](http://math.stackexchange.com/questions/1439148/a-generalization-of-thomsen-theorem-and-tucker-circle). Now, I give a more generalization of these theorems as following:

Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, $L$ be a line on the plane. Let $L$ meets $A_1A_2$, $A_2A_3$, $A_3A_4$, $A_4A_5$, $A_5A_6$, $A_6A_1$ at $B_2$, $B_3$, $B_4$, $B_5$, $B_6$, $B_1$ respectively.

**Problem:** Let $C_1$ be a point on the line $A_1A_4$. Let $C_1B_2$ meets $A_2A_5$ at $C_2$. Let $C_2B_3$ meets $A_3A_6$ at $C_3$. Let $C_3B_4$ meets $A_1A_4$ at $C_4$. Let $C_4B_5$ meets $B_2B_5$ at $C_5$. Let $C_5B_6$ meets $A_3A_6$ at $C_6$. Let $C_6B_1$ meets $A_1A_4$ at $C_7$. Then: 

1. Six points $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ lie on a conic if only if six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ lie on a conic.

2. $C_7 \equiv C_1$ if only if six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ lie on a conic.

**Remark:** When the conic through $A_1, A_2, A_3, A_4, A_5, A_6$ is the circumcircle and $L$ at infinity, the item 1 is the [Tucker circle theorem](http://mathworld.wolfram.com/TuckerCircles.html). When the conic through $A_1, A_2, A_3, A_4, A_5, A_6$ is the Steiner inellipse and $L$ at infinity, item 2 is the [Thomsen theorem](https://en.wikipedia.org/wiki/Thomsen%27s_theorem).

> **My question:** can you give a proof for the problem?

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/h9Cur.png