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centroids of triangles, tetrahedron - 1998 Romania NMO VII p4

Source:

August 14, 2024

geometry3D geometrytetrahedron

Problem Statement

Let

ABCD

be an arbitrary tetrahedron. The bisectors of the angles

\angle BDC

,

N

and

\angle ADB

intersect

BC

,

CA

and

AB

, in the points

M

,

N

,

P

, respectively.
a) Show that the planes

(ADM)

,

(BDN)

and

(CDP)

have a common line

d

.
b) Let the points

A' \in (AD)

,

B' \in (BD)

and

C' \in (CD)

be such that

(AA') = (BB') = (CC')

; show that if

G

and

G'

are the centroids of

ABC

and

A'B'C'

then the lines

GG'

and

d

are either parallel or identical.

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