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6

Part of 2004 Oral Moscow Geometry Olympiad

Problems(2)

principal diagonal in convex hexagon <= 2 /\sqrt3

Source: 2004 Oral Moscow Geometry Olympiad grade 9 p6

10/27/2020
The length of each side and each non-principal diagonal of a convex hexagon does not exceed 11. Prove that this hexagon contains a principal diagonal whose length does not exceed 23\frac{2}{\sqrt3}.
geometrygeometric inequalitydiagonalhexagon
computational in tetrahedron DABC, <ACB = <ADB, (CD) _|_ (ABC)

Source: 2004 Oral Moscow Geometry Olympiad grade 10 p6

10/28/2020
In the tetrahedron DABCDABC : ∠ACB=∠ADB\angle ACB = \angle ADB, (CD)⊥(ABC)(CD) \perp (ABC). In triangle ABCABC, the altitude hh drawn to the side ABAB and the distance dd from the center of the circumscribed circle to this side are given. Find the length of the CDCD.
geometry3D geometrytetrahedron