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Taiwan Mathematics Olympiad
2024 Taiwan Mathematics Olympiad
4
4
Part of
2024 Taiwan Mathematics Olympiad
Problems
(1)
Easy Taiwanese Geometry
Source: 2024 Taiwan Mathematics Olympiad
1/31/2024
Suppose
O
O
O
is the circumcenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
, and
E
,
F
E, F
E
,
F
are points on segments
C
A
CA
C
A
and
A
B
AB
A
B
respectively with
E
,
F
≠
A
E, F \neq A
E
,
F
=
A
. Let
P
P
P
be a point such that
P
B
=
P
F
PB = PF
PB
=
PF
and
P
C
=
P
E
PC = PE
PC
=
PE
. Let
O
P
OP
OP
intersect
C
A
CA
C
A
and
A
B
AB
A
B
at points
Q
Q
Q
and
R
R
R
respectively. Let the line passing through
P
P
P
and perpendicular to
E
F
EF
EF
intersect
C
A
CA
C
A
and
A
B
AB
A
B
at points
S
S
S
and
T
T
T
respectively. Prove that points
Q
,
R
,
S
Q, R, S
Q
,
R
,
S
, and
T
T
T
are concyclic.Proposed by Li4 and usjl
Taiwan
geometry