geometry problems from Team Selection Tests (TST) from Georgia with aops links in the names
2005
In triangle $ ABC$ we have $ \angle{ACB} = 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD = AC$ and $ DB = DC$. Prove that $ \angle{BAC} = 3\angle{BAD}$.
Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ=QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}=\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.
On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.
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