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China Contests
China National Olympiad
2019 China National Olympiad
3
3
Part of
2019 China National Olympiad
Problems
(1)
Prove line tangent to circle
Source: China Mathematical Olympiad 2019 Q3
11/14/2018
Let
O
O
O
be the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
(
A
B
<
A
C
AB<AC
A
B
<
A
C
), and
D
D
D
be a point on the internal angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
. Point
E
E
E
lies on
B
C
BC
BC
, satisfying
O
E
∥
A
D
OE\parallel AD
OE
∥
A
D
,
D
E
⊥
B
C
DE\perp BC
D
E
⊥
BC
. Point
K
K
K
lies on
E
B
EB
EB
extended such that
E
K
=
E
A
EK=EA
E
K
=
E
A
. The circumcircle of
△
A
D
K
\triangle ADK
△
A
DK
meets
B
C
BC
BC
at
P
≠
K
P\neq K
P
=
K
, and meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
Q
≠
A
Q\neq A
Q
=
A
. Prove that
P
Q
PQ
PQ
is tangent to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
.
geometry
circumcircle