MathDB

(030) 4

Part of 1982 Tournament Of Towns

Problems(1)

TOT 030 1982 Autumn S4 vector sum in a regular n-gon / tetrahedron

Source:

8/18/2019
(a) K1,K2,...,KnK_1,K_2,..., K_n are the feet of the perpendiculars from an arbitrary point MM inside a given regular nn-gon to its sides (or sides produced). Prove that the sum MK1→+MK2→+...+MKn→\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n} equals n2MO→\frac{n}{2}\overrightarrow{MO}, where OO is the centre of the nn-gon.
(b) Prove that the sum of the vectors whose origin is an arbitrary point MM inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from MM to the faces of the tetrahedron equals 43MO→\frac43 \overrightarrow{MO}, where OO is the centre of the tetrahedron.
(VV Prasolov, Moscow)
geometry3D geometrytetrahedronvectorregular polygonregular tetrahedron