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

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August 18, 2019
geometry3D geometrytetrahedronvectorregular polygonregular tetrahedron

Problem Statement

(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)
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