# A new theorem (discovered in 2013) equivalent to Brianchon theorem (the old theorem) discovered in XIX century?

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

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