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Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
9
9
Part of
2019 Sharygin Geometry Olympiad
Problems
(1)
Sharygin CR 2019 P9
Source:
3/6/2019
Let
A
M
A_M
A
M
be the midpoint of side
B
C
BC
BC
of an acute-angled
Δ
A
B
C
\Delta ABC
Δ
A
BC
, and
A
H
A_H
A
H
be the foot of the altitude to this side. Points
B
M
,
B
H
,
C
M
,
C
H
B_M, B_H, C_M, C_H
B
M
,
B
H
,
C
M
,
C
H
are defined similarly. Prove that one of the ratios
A
M
A
H
:
A
H
A
,
B
M
B
H
:
B
H
B
,
C
M
C
H
:
C
H
C
A_MA_H : A_HA, B_MB_H : B_HB, C_MC_H : C_HC
A
M
A
H
:
A
H
A
,
B
M
B
H
:
B
H
B
,
C
M
C
H
:
C
H
C
is equal to the sum of two remaining ratios
geometry