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4

Part of 2021 Oral Moscow Geometry Olympiad

Problems(2)

three lines throught vertices and circumcenters are concurrent

Source: 2021 Oral Moscow Geometry Olympiad grades 8-9 p4

5/17/2021
On the diagonal ACAC of cyclic quadrilateral ABCDABCD a point EE is chosen such that ∠ABE=∠CBD\angle ABE = \angle CBD. Points O,O1,O2O,O_1,O_2 are the circumcircles of triangles ABC,ABEABC, ABE and CBECBE respectively. Prove that lines DO,AO1DO,AO_{1} and CO2CO_{2} are concurrent.
geometrycyclic quadrilateralcircumcircle
Sphere touches all edges of an octahedron STABCD. Prove that ABCD are complanar

Source: 2021 Oral Moscow Geometry Olympiad grades 10-11 p4

5/17/2021
Points STABCDSTABCD in space form a convex octahedron with faces SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDASAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA such that there exists a sphere that is tangent to all of its edges. Prove that A,B,C,DA,B,C,D lie in one plane.
geometry3D geometryoctahedronsphere