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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2021 Sharygin Geometry Olympiad
3
3
Part of
2021 Sharygin Geometry Olympiad
Problems
(1)
Prove the concurrency of three lines
Source: XVII Sharygin Correspondence Round P3
3/2/2021
Altitudes
A
A
1
,
C
C
1
AA_1,CC_1
A
A
1
,
C
C
1
of acute-angles
A
B
C
ABC
A
BC
meet at point
H
H
H
;
B
0
B_0
B
0
is the midpoint of
A
C
AC
A
C
. A line passing through
B
B
B
and parallel to
A
C
AC
A
C
meets
B
0
A
1
,
B
0
C
1
B_0A_1 , B_0C_1
B
0
A
1
,
B
0
C
1
at points
A
′
,
C
′
A',C'
A
′
,
C
′
respectively. Prove that
A
A
′
,
C
C
′
AA',CC'
A
A
′
,
C
C
′
and
B
H
BH
B
H
concur.
geometry