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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
8
8
Part of
2024 China Team Selection Test
Problems
(1)
Golden ratio
Source: 2024 CTST P8
3/11/2024
In
△
A
B
C
\triangle {ABC}
△
A
BC
, tangents of the circumcircle
⊙
O
\odot {O}
⊙
O
at
B
,
C
B, C
B
,
C
and at
A
,
B
A, B
A
,
B
intersects at
X
,
Y
X, Y
X
,
Y
respectively.
A
X
AX
A
X
cuts
B
C
BC
BC
at
D
{D}
D
and
C
Y
CY
C
Y
cuts
A
B
AB
A
B
at
F
{F}
F
. Ray
D
F
DF
D
F
cuts arc
A
B
AB
A
B
of the circumcircle at
P
{P}
P
.
Q
,
R
Q, R
Q
,
R
are on segments
A
B
,
A
C
AB, AC
A
B
,
A
C
such that
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
are collinear and
Q
R
∥
B
O
QR \parallel BO
QR
∥
BO
. If
P
Q
2
=
P
R
⋅
Q
R
PQ^2=PR \cdot QR
P
Q
2
=
PR
⋅
QR
, find
∠
A
C
B
\angle ACB
∠
A
CB
.
geometry
ratio