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Contests
National and Regional Contests
Korea Contests
Korea Winter Program Practice Test
2020 Korean MO winter camp
#5
#5
Part of
2020 Korean MO winter camp
Problems
(1)
Two tangent circles
Source: 2020 Korean MO winter camp Test 1 P5
9/7/2020
□
A
B
C
D
\square ABCD
□
A
BC
D
is a quadrilateral with
∠
A
=
2
∠
C
<
9
0
∘
\angle A=2\angle C <90^\circ
∠
A
=
2∠
C
<
9
0
∘
.
I
I
I
is the incenter of
△
B
A
D
\triangle BAD
△
B
A
D
, and the line passing
I
I
I
and perpendicular to
A
I
AI
A
I
meets rays
C
B
CB
CB
and
C
D
CD
C
D
at
E
,
F
E,F
E
,
F
respectively. Denote
O
O
O
as the circumcenter of
△
C
E
F
\triangle CEF
△
CEF
. The line passing
E
E
E
and perpendicular to
O
E
OE
OE
meets ray
O
F
OF
OF
at
Q
Q
Q
, and the line passing
F
F
F
and perpendicular to
O
F
OF
OF
meets ray
O
E
OE
OE
at
P
P
P
. Prove that the circle with diameter
P
Q
PQ
PQ
is tangent to the circumcircle of
△
B
C
D
\triangle BCD
△
BC
D
.
geometry