2012 ToT Fall Junior A p6 centre of mass of 2n-gon
Source:
March 22, 2020
geometrycombinatorial geometryperpendicularCentroidregular polygon
Problem Statement
(a) A point is marked inside a circle. Two perpendicular lines drawn through intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines.
(b) A regular -gon () with centre is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from to the vertices of the -gon mark points on the circle. Then the -gon is rotated about . The rays going from to the new locations of vertices mark new points on the circle. Let and be the centres of gravity of old and new points respectively. Prove that .