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National and Regional Contests
China Contests
China National Olympiad
2017 China National Olympiad
2
2
Part of
2017 China National Olympiad
Problems
(1)
Concyclic iff collinear
Source: China Mathematical Olympiad 2017 Q2
11/23/2016
In acute triangle
A
B
C
ABC
A
BC
, let
⊙
O
\odot O
⊙
O
be its circumcircle,
⊙
I
\odot I
⊙
I
be its incircle. Tangents at
B
,
C
B,C
B
,
C
to
⊙
O
\odot O
⊙
O
meet at
L
L
L
,
⊙
I
\odot I
⊙
I
touches
B
C
BC
BC
at
D
D
D
.
A
Y
AY
A
Y
is perpendicular to
B
C
BC
BC
at
Y
Y
Y
,
A
O
AO
A
O
meets
B
C
BC
BC
at
X
X
X
, and
O
I
OI
O
I
meets
⊙
O
\odot O
⊙
O
at
P
,
Q
P,Q
P
,
Q
. Prove that
P
,
Q
,
X
,
Y
P,Q,X,Y
P
,
Q
,
X
,
Y
are concyclic if and only if
A
,
D
,
L
A,D,L
A
,
D
,
L
are collinear.
geometry