5

Part of 2006 Oral Moscow Geometry Olympiad

Problems(2)

midpoint wanted, equilateral triangles on sides of a triangle related

Source: 2006 Oral Moscow Geometry Olympiad grades 8-9 p5

Equilateral triangles

ABC_1, BCA_1, CAB_1

are built on the sides of the triangle

ABC

to the outside. On the segment

A_1B_1

to the outer side of the triangle

A_1B_1C_1

, an equilateral triangle

A_1B_1C_2

is constructed. Prove that

C

is the midpoint of the segment

C_1C_2

.
(A. Zaslavsky)

geometrymidpointEquilateral

exists cyclic quadrilateral, section of a pyramid with convex quadr. base?

Source: 2006 Oral Moscow Geometry Olympiad grades 10-11 p5

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle?
(M. Volchkevich)

geometrycyclic quadrilateral3D geometrypyramid