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National and Regional Contests
Thailand Contests
Thailand National Olympiad
2023 Thailand Mathematical Olympiad
8
8
Part of
2023 Thailand Mathematical Olympiad
Problems
(1)
Intersection of tangent at A,B config
Source: 2023 Thailand MO Day 2 P8
5/12/2023
Let
A
B
C
ABC
A
BC
be an acute triangle. The tangent at
A
,
B
A,B
A
,
B
of the circumcircle of
A
B
C
ABC
A
BC
intersect at
T
T
T
. Line
C
T
CT
CT
meets side
A
B
AB
A
B
at
D
D
D
. Denote by
Γ
1
,
Γ
2
\Gamma_1,\Gamma_2
Γ
1
,
Γ
2
the circumcircle of triangle
C
A
D
CAD
C
A
D
, and the circumcircle of triangle
C
B
D
CBD
CB
D
, respectively. Let line
T
A
TA
T
A
meet
Γ
1
\Gamma_1
Γ
1
again at
E
E
E
and line
T
B
TB
TB
meet
Γ
2
\Gamma_2
Γ
2
again at
F
F
F
. Line
E
F
EF
EF
intersects sides
A
C
,
B
C
AC,BC
A
C
,
BC
at
P
,
Q
P,Q
P
,
Q
, respectively. Prove that
E
F
=
P
Q
+
A
B
EF=PQ+AB
EF
=
PQ
+
A
B
.
geometry
circumcircle