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Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 3
2020 Taiwan TST Round 3
1
1
Part of
2020 Taiwan TST Round 3
Problems
(1)
Easy Geometry in Taiwan TST
Source: 2020 Taiwan TST Round 3
5/22/2020
Let
Ω
\Omega
Ω
be the
A
A
A
-excircle of triangle
A
B
C
ABC
A
BC
, and suppose that
Ω
\Omega
Ω
is tangent to lines
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at points
D
D
D
,
E
E
E
, and
F
F
F
, respectively. Let
M
M
M
be the midpoint of segment
E
F
EF
EF
. Two more points
P
P
P
and
Q
Q
Q
are on
Ω
\Omega
Ω
such that
E
P
EP
EP
and
F
Q
FQ
FQ
are both parallel to
D
M
DM
D
M
. Let
B
P
BP
BP
meet
C
Q
CQ
CQ
at point
X
X
X
. Prove that the line
A
M
AM
A
M
is the angle bisector of
∠
X
A
D
\angle XAD
∠
X
A
D
. Proposed by Shuang-Yen Lee
geometry
excircle