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

Since 2013, I found a very nice configuration: $$N$$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization of Pascal theorem even with $$1$$st-chain of six circles). I am looking for a proof of one of these following problems:

Let $$(C_{1})$$, $$(C_{2})$$, $$(C_{3})$$ be three circles in a plane. Let $$A_{i1}, A_{i4}$$ be points lie on circle $$(C_{1})$$; $$A_{i2}, A_{i5}$$ be points lie on circle $$(C_{2})$$; $$A_{i3}, A_{i6}$$ be points lie on circle $$(C_{3})$$ for $$i=1, 2, \cdots n$$.

So that six points $$A_{11}$$, $$A_{12}$$ , $$A_{13}$$, $$A_{14}$$, $$A_{15}$$, $$A_{16}$$, lie on a circle and four poins $$A_{ij}$$, $$A_{i{j+1}}$$, $$A_{{i+1}{j+1}}$$, $$A_{{i+1}{j}}$$ lie on a circle for $$i=1$$, $$2$$, $$\cdots$$, $$n$$ and $$j=1, 2, \cdots, 5$$

Problem 1.(Already has a proof in here) Four points $$A_{i5}, A_{i6}, A_{{(i+1)}5}, A_{{(i+1)}6}$$ lie on a circle and six points $$A_{i1}$$, $$A_{i2}$$, $$A_{i3}$$, $$A_{i4}$$, $$A_{i5}$$, $$A_{i6}$$ lie on a circle for $$i=1$$, $$2$$, $$\cdots$$, $$n$$.

Denote $$(O_{ij})$$ is the circle through $$A_{ij}$$, $$A_{i{j+1}}$$, $$A_{{i+1}{j+1}}$$, $$A_{{i+1}{j}}$$ for $$i=1$$, $$2$$, $$\cdots$$, $$n$$ and $$j=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$

Problem 2. Three lines $$O_{i1}O_{k4}$$, $$O_{i2}O_{k5}$$, $$O_{i3}O_{k6}$$ are concurrent for all $$i, k =1, 2, \cdots$$

Problem 3. Three lines $$O_{i1}O_{k1}$$, $$O_{i3}O_{k3}$$, $$O_{i5}O_{k5}$$ are concurrent, for all $$i, k =1, 2, \cdots$$

Problem 4. Three lines $$O_{i2}O_{k2}$$, $$O_{i4}O_{k4}$$, $$O_{i6}O_{k6}$$ are concurrent for all $$i, k =1, 2, \cdots$$

Problem 5. If $$(O_{i1})$$ meets $$(O_{k1})$$ at wo points $$P_1$$, $$P'_1$$; $$(O_{i2})$$ meets $$(O_{k2})$$ at wo points $$P_2$$, $$P'_2$$, $$\cdots$$, similarly $$(O_{i6})$$ meets $$(O_{k6})$$ at wo points $$P_6$$, $$P'_6$$ then $$12$$ points $$P_1, \cdots, P_6, P'_1, \cdots, P'_6$$ lie on a circle.

Problem 6. If $$(O_{i1})$$ meets $$(O_{k4})$$ at wo points $$P_1$$, $$P_4$$, $$(O_{i2})$$ meets $$(O_{k5})$$ at wo points $$P_2, P_5$$, $$(O_{i3})$$ meets $$(O_{k6})$$ at wo points $$P_3$$, $$P_6$$. Then six the six points lie on a circle $$P_1,\cdots, P_6$$ lie on a circle. $$i = 1, 2, \cdots$$

Note: The case $$1$$-st. Dao Thanh Oai, Problem 3845, Crux mathematicorum, issue May, 2013

See also:

• Is "7. More and more properties" part of the question? And what is the status of the stuff that comes after it ("2-The case 1st...")? – მამუკა ჯიბლაძე Dec 16 '18 at 3:25
• I mean there are many peroperties of this configuration. – Đào Thanh Oai Dec 16 '18 at 3:30
• I agree, and it is beautiful, but you should separate out a clearly stated question. Your post is meant to be a question. – მამუკა ჯიბლაძე Dec 16 '18 at 3:32
• @მამუკაჯიბლაძე I deleted 7. – Đào Thanh Oai Dec 16 '18 at 3:34
• I don't think you should. It was completely OK to inform the reader that this configuration has a lot more properties. What is not OK (for me) is that by the time I finished reading, I could not figure out what exactly do you ask. Presumably how to prove that certain six points lie on a circle and certain other four points lie on another circle? Why don't you clearly formulate this? – მამუკა ჯიბლაძე Dec 16 '18 at 3:37