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The centroid of A"B"C"D" is the same as the centroid of ABCD

Source: Canada National Mathematical Olympiad 1982 - Problem 5

September 30, 2011

geometry3D geometrytetrahedronvectorgeometry proposed

Problem Statement

The altitudes of a tetrahedron

ABCD

are extended externally to points

A'

B'

C'

, and

D'

, where

AA' = k/h_a

BB' = k/h_b

CC' = k/h_c

, and

DD' = k/h_d

. Here,

k

is a constant and

h_a

denotes the length of the altitude of

ABCD

from vertex

A

, etc. Prove that the centroid of tetrahedron

A'B'C'D'

coincides with the centroid of

ABCD