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Balkan MO Shortlist
2007 Balkan MO Shortlist
G4
G4
Part of
2007 Balkan MO Shortlist
Problems
(1)
h<=2H , altitudes inequality of an acute triangle inscribed in a triangle
Source: 2007 Balkan Shortlist BMO G4 - Bulgaria
4/5/2020
Points
M
,
N
M,N
M
,
N
and
P
P
P
on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
of
△
A
B
C
\vartriangle ABC
△
A
BC
are such that
△
M
N
P
\vartriangle MNP
△
MNP
is acute. Denote by
h
h
h
and
H
H
H
the lengths of the shortest altitude of
△
A
B
C
\vartriangle ABC
△
A
BC
and the longest altitude of
△
M
N
P
\vartriangle MNP
△
MNP
. Prove that
h
≤
2
H
h \le 2H
h
≤
2
H
.
geometric inequality
altitudes
altitude
geometry