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Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2023 Abelkonkurransen Finale
1a
1a
Part of
2023 Abelkonkurransen Finale
Problems
(1)
Show that X,Y, Z are the midpoints of sides of ABC [2023 Abel, 1a]
Source:
3/12/2024
In the triangle
A
B
C
ABC
A
BC
,
X
X
X
lies on the side
B
C
BC
BC
,
Y
Y
Y
on the side
C
A
CA
C
A
, and
Z
Z
Z
on the side
A
B
AB
A
B
with
Y
X
∥
A
B
,
Z
Y
∥
B
C
YX \| AB, ZY \| BC
Y
X
∥
A
B
,
Z
Y
∥
BC
, and
X
Z
∥
C
A
XZ \| CA
XZ
∥
C
A
. Show that
X
,
Y
X,Y
X
,
Y
, and
Z
Z
Z
are the midpoints of the respective sides of
A
B
C
ABC
A
BC
.
geometry