Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.
Problem
Source: China TST 2004 Quiz
Tags: geometry unsolved, geometry
02.02.2009 02:06
This solution is not very nature but I hope it is OK Denote by $ F'$ the intersection of $ AE$ with the semi-circle with diameter $ AC$ erected outside triangle $ ABC$. We have $ \angle F'ED=\angle F'AC=\angle F'HC$, it implies that quadrilateral $ F'EDH$ is cyclic. Therefore, $ \angle F'DH=\angle F'EH=\angle ABH$, so we conclude $ DF'$ is parallel to $ AB$, i.e., $ F\equiv F'$. The conclusion follows.
02.02.2009 08:25
Dear April and Mathlinkers, your nice proof is in a synthetic point of view, the little Pappus's theorem. I saw this problem on Mathlinks... where? Sincerely Jean-Louis
02.02.2009 09:16
This problem is special case of USA 2005 Problem 3 http://www.mathlinks.ro/viewtopic.php?p=213011#213011
03.01.2020 12:47