# Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper

In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many topics in mathematics.

The calculation of barycentric coordinate for concurrence given by N. Dergiades takes more than 72 pages A4. In 09-2014, N. Dergiades gave an elegant proof of this theorem and renamed this theorem: Dao’s theorem on six circumcenters associated with a cyclic hexagon. In 10-2014, T. Cohl, a Taiwan student, gave a synthetic proof for this theorem. Two proofs were published in the Forum Geometricorum journal.

We consider the following configuration: Let $L_1, L_2, L_3, L_4, L_5, L_6$ be six lines and let $P_{ij}= L_i \cap Lj$, such that $P_{12},P_{23}, P_{34}, P_{45}, P_{56}, P_{61}$ lie on a circle. Let $(O_{ijk})$ is circle $(P_{ij}, P_{jk}, P_{ik})$ with center $O_{ijk}$. Let $(O_{ijk})$ meets $(O_{jkh})$ again at $P'_{jk}$. (We taking subscripts modulo 6.)

I am looking a solution for the problem as follows:

Problem 1 Ngo Quang Duong: Let $P_{12}P_{45}, P_{34}P_{61}, P_{56}P_{23}$ are concurrent at $P$. Then six points $P'_{12}$, $P'_{23}$, $P'_{34}$, $P'_{45}$, $P'_{56}$, $P'_{61}$ lie on a circle.

PS: I posted this topic because the problem 1 look like Miquel's Pentagram Theorem

• Would O.T. Dao be the OP? If so, from one mathematician to another: Don't refer to your own theorems as `remarkable', even if they are! – Jon Bannon Mar 19 '16 at 17:52
• @Jon, cut her some slack. Most foreign speakers aren't familiar with linguistic etiquette in English--how to assume the proper tone. "I published a theorem illustrating the remarkable / beautiful connections between ..." would be perfectly acceptable. – Tom Copeland Mar 19 '16 at 18:16
• @Tom: You are right! The wording you suggest is just fine. Thanks for providing a positive suggestion! – Jon Bannon Mar 19 '16 at 21:09
• @Jon, I'm sure that's exactly the sentiment the OP intended to convey. – Tom Copeland Mar 19 '16 at 23:36
• @OaiThanhĐào : one never says "I thank to you..."; it's just "Thank you...". – Dima Pasechnik Apr 17 '16 at 16:18