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A property of the Euler-Poncelet Point / Một tính chất của điểm Euler-Poncelet
(Lướt xuống sẽ thấy phần tiếng Việt, hình được đính kèm ở cuối blog) -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- A property of the Euler-Poncelet Point An apprentice of mine (Dương Phú Tài) sent me this corollary from a problem composed by Lê Trung Hiếu: Given a complete quadrilateral $ABCD.EF$ with $ABCD$ being concyclic, define $P$ as the Euler-Poncelet point. $O$, $H$ respectively are the circumcenter and orthocenter of $\Delta AEC$. $O'$, $H'$ are defined similarly for $\Delta AFC$. a) Prove that the midpoint of segment $AC$ lies on the Gauss-Newton line, defined as line $\omega$ of quadrilateral $HOH'O'$ b) Prove that $P$ lies on $\omega$ - Euler-Poncelet Point: The intersection of the Euler Circles of $\Delta ABC, ABD, ACD, BCD$ Although this problem could be stated more clearly, it is nonetheless an interesting issue. The Euler-Poncelet point is not a very explored problem in synthetic geometry, so I hope this prop...
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