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International Contests
Tuymaada Olympiad
2016 Tuymaada Olympiad
3
3
Part of
2016 Tuymaada Olympiad
Problems
(1)
Concurrent
Source: Tuymaada 2016, Seniors/P3
7/22/2016
Altitudes
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
of an acute triangle
A
B
C
ABC
A
BC
meet at
H
H
H
.
A
0
A_0
A
0
,
B
0
B_0
B
0
,
C
0
C_0
C
0
are the midpoints of
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively. Points
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
on the segments
A
H
AH
A
H
,
B
H
BH
B
H
,
H
C
1
HC_1
H
C
1
respectively are such that
∠
A
0
B
2
A
2
=
∠
B
0
C
2
B
2
=
∠
C
0
A
2
C
2
=
9
0
∘
\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ
∠
A
0
B
2
A
2
=
∠
B
0
C
2
B
2
=
∠
C
0
A
2
C
2
=
9
0
∘
. Prove that the lines
A
C
2
AC_2
A
C
2
,
B
A
2
BA_2
B
A
2
,
C
B
2
CB_2
C
B
2
are concurrent.
geometry