In 2013, I found a new problem as follows: Let six points $A_1$, $A_2$, ...$A_6$ lie on a circle $(O_1)$, and the six points $B_1$, $B_2$,...,$B_6$ lie on another circle $(O_2)$. If the quadruples $A_i$, $A_{i+1}$, $B_{i+1}$, $B_i$ lie on a circles with centers $C_i$ for $i=1,2,...,5$ then $A_6$, $A_1$, $B_1$, $B_6$ lie on a circle (namely center of the new circle is $C_6$) and $C_1C_4$, $C_2C_5$, $C_3C_6$ are concurrent.

The proof [3][4] show that the hexagon $C_1C_2C_3C_4C_5C_6$ circumscribed around a conic section with two Focus are $O_1$, $O_2$

The Theorem was re-discovered by Szilasis in 2017 (see [5])

**Converse of theorem:** Let a hexagon $C_1C_2C_3C_4C_5C_6$ which $C_1C_4, C_2C_5, C_3C_6$ are concurrent then exist many configuration of eight circles as figuration above.

If the converse theorem is true, so the theorem is equivalent with an old theorem due to Brianchon

**Special case:** Let a hexagon $C_1C_2C_3C_4C_5C_6$ circumscribed a conic with two focus $O_1$, $O_2$. Draw six circles $(C_1)$, $(C_2)$,....,$(C_6)$ with centers $C_1$, $C_2$,...$C_6$ and radii $C_1O_1$, $C_2O_1$, ...., $C_6O_1$ respectively. Let circles $(C_i)$ meets circle $(C_{i+1})$ again at $A_i$ then six points $A_1, A_2,..., A_6$ lie on a circle, namely circle $(A)$. Similarly: Draw six circles $(C_1)$,$(C_2)$,....,$(C_6)$ with centers $C_1$, $C_2$,...$C_6$ and radii $C_1O_2$, $C_2O_2$, ...., $C_6O_2$ respectively. Let circles $(C_i)$ meets circle $(C_{i+1})$ again at $B_i$ then six points $B_1, B_2,..., B_6$ lie on a circle, namely the circle $(B)$.

Question 1:Is the converse of theorem true?

Question 2:How can one prove the radius of circle $(A)$ is equal to the radius of circle $(B)$?

**The dual of theorem on the ball creat a beautifull basket (see theorem dual in [8])**

**Some relation problem**

Dao's theorem on six circumcenters

Some Problems On Apollonian Gasket

Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions?

A chain of six circles associated with six points on a circle (in Mobius plane)

$N$-$th$ closed chain of six circles

**Reference:**

1 - Dao, O.T.: Problem 3845, Crux Mathematicorum, 39, Issue May 2013

2-https://www.geogebra.org/material/show/id/Zk3F5y5X

3 - J. Chris Fisher, Problem 3945, Crux Mathematicorum, Volume 40, Issue May, 2014

[4]-Michel Bataille, Solution to Problem 3945, Crux Mathematicorum, Volume 41, Issue May, 2015

[5]-Gábor Gévay, A remarkable theorem on eight circles, Forum Geométrico rum, Volume 18 (2018), 401--408

[6]-Ákos G.Horváth, A note on the centers of a closed chain of circles