5

Part of 2018 Oral Moscow Geometry Olympiad

Problems(2)

locus of midpoints of altitudes wanted, fixed circle and point given

Source: 2018 Oral Moscow Geometry Olympiad grades 8-9 p5

The circle circumscribed about an acute triangle

ABC

C

are fixed. Orthocenter

H

moves in a circle with center at point

C

. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices

A

B

geometryLocusfixedmidpointsorthocenter

ants on surface a tetrahedron, distance inequality

Source: 2018 Oral Moscow Geometry Olympiad grades 10-11 p5

Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.

geometry3D geometrytetrahedroninequalitiesgeometric inequality