MathDB

5

Part of 2015 Oral Moscow Geometry Olympiad

Problems(2)

rhombus BCDE ouside equilateral ABE, AF <BD wanted

Source: Oral Moscow Geometry Olympiad 2015 grades 8-9 p5

8/8/2019
On the BEBE side of a regular ABEABE triangle, a BCDEBCDE rhombus is built outside it. The segments ACAC and BDBD intersect at point FF. Prove that AF<BDAF <BD.
geometric inequalityrhombusEquilateralgeometry
prove that planes are tangent to a fixed sphere or pass through a fixed point

Source: Oral Moscow Geometry Olympiad 2015 grades 10-11 p5

8/6/2019
A triangle ABCABC and spheres are given in space S1S_1 and S2S_2, each of which passes through points A,BA, B and CC. For points MM spheres S1S_1 not lying in the plane of triangle ABCABC are drawn lines MA,MBMA, MB and MCMC, intersecting the sphere S2S_2 for the second time at points A1,B1A_1,B_1 and C1C_1, respectively. Prove that the planes passing through points A1,B1A_1, B_1 and C1C_1, touch a fixed sphere or pass through a fixed point.
geometry3D geometrysphereFixed pointplanesMoscow