# Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems:

Let three $$(A)$$, $$(B)$$, $$(C)$$ circle (cyan circles) are tangent together and tangent with $$(G)$$ and $$(K)$$. Two circles $$(G)$$, $$K$$ are Soddy circles of triangle $$\triangle ABC$$

Construct three circle $$(M_{11})$$, $$(M_{21})$$, $$(M_{31})$$ (blue)

• $$(M_{11})$$ is tangent $$(B)$$ and $$(C)$$ and $$(G)$$ (or $$(K)$$)

• $$(M_{21})$$ is tangent $$(A)$$ and $$(C)$$ and $$(G)$$ (or $$(K)$$)

• $$(M_{31})$$ is tangent $$(A)$$ and $$(B)$$ and $$(G)$$ (or $$(K)$$)

Continuing construct three circles $$(M_{1i})$$, $$(M_{2i})$$, $$(M_{3i})$$ so that:

• $$(M_{1i})$$ is tangent $$(B)$$ and $$(C)$$ at $$P_{1ib}$$, $$P_{1ic}$$ and $$(M_{1i})$$ tangent to $$(M_{1(i-1)})$$ at $$P_{1{i-1}}$$

• $$(M_{2i})$$ is tangent $$(A)$$ and $$(C)$$ at $$P_{2ia}$$, $$P_{2ic}$$ and $$(M_{2i})$$ tangent to $$(M_{2(i-1)})$$ at $$P_{2{i-1}}$$

• $$(M_{3i})$$ is tangent $$(A)$$ and $$(B)$$ at $$P_{3ia}$$, $$P_{3ic}$$ and $$(M_{3i})$$ tangent to $$(M_{3(i-1)})$$ at $$P_{3{i-1}}$$

For $$i=2, 3, ...n$$

Denote:

• $$(H_i)$$ is circle through tangent point $$(M_{ki})$$ and $$(M_{k(i-1)})$$ for $$k=1$$, $$2$$, $$3$$

Problem 1: Three lines $$AM_{1i}$$, $$BM_{2i}$$, $$CM_{3i}$$, $$P_{2ia}$$, $$P_{2ic}$$, $$P_{3ia}$$, $$P_{3ic}$$ are concurrent; denote the points of concurrence is $$K_i$$ for $$i=1, 2, ..., n$$

Problem 2: Three lines $$M_{1k}M_{1i}$$, $$M_{2k}M_{2i}$$, $$M_{3k}M_{3i}$$ are concurrent, denote the points of concurrence is $$H_{ik}$$ for $$i, k=1, 2, ..., n$$

Problem 3: Six point points $$P_{1ib}$$, $$P_{1ic}$$, $$P_{2ia}$$, $$P_{2ic}$$, $$P_{3ia}$$, $$P_{3ic}$$ are concyclic, denote $$O_i$$ is the circumcenter of this circle.

Problem 4: These points $$K_i$$, $$H_{ik}$$, $$(O_i)$$ lie on a line for $$i, k=1, 2, ..., n$$

More properties in here

• @Martin Sleziak Thank you very much for your many edit – Đào Thanh Oai Dec 16 '18 at 15:35