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Vietnam Contests
Vietnam Team Selection Test
2023 Vietnam Team Selection Test
3
3
Part of
2023 Vietnam Team Selection Test
Problems
(1)
Two tangents line concur on MK
Source: Vietnam TST 2023 P3
4/13/2023
Let
A
B
C
ABC
A
BC
be an acute, non-isosceles triangle with circumcircle
(
O
)
(O)
(
O
)
.
B
E
,
C
F
BE, CF
BE
,
CF
are the heights of
△
A
B
C
\triangle ABC
△
A
BC
, and
B
E
,
C
F
BE, CF
BE
,
CF
intersect at
H
H
H
. Let
M
M
M
be the midpoint of
A
H
AH
A
H
, and
K
K
K
be the point on
E
F
EF
EF
such that
H
K
⊥
E
F
HK \perp EF
HK
⊥
EF
. A line not going through
A
A
A
and parallel to
B
C
BC
BC
intersects the minor arc
A
B
AB
A
B
and
A
C
AC
A
C
of
(
O
)
(O)
(
O
)
at
P
P
P
,
Q
Q
Q
, respectively.Show that the tangent line of
(
C
Q
E
)
(CQE)
(
CQE
)
at
E
E
E
, the tangent line of
(
B
P
F
)
(BPF)
(
BPF
)
at
F
F
F
, and
M
K
MK
M
K
concur.
geometry