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Polish MO Finals
2022 Polish MO Finals
1
1
Part of
2022 Polish MO Finals
Problems
(1)
Lots of perpendiculars
Source: Poland 73-3-1
3/31/2022
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
. The angle bisector of
B
A
C
BAC
B
A
C
intersects the side
B
C
BC
BC
and the circumcircle of
A
B
C
ABC
A
BC
at
D
D
D
and
M
≠
A
M\neq A
M
=
A
, respectively. Points
X
X
X
and
Y
Y
Y
are chosen so that
M
X
⊥
A
B
MX \perp AB
MX
⊥
A
B
,
B
X
⊥
M
B
BX \perp MB
BX
⊥
MB
,
M
Y
⊥
A
C
MY \perp AC
M
Y
⊥
A
C
, and
C
Y
⊥
M
C
CY \perp MC
C
Y
⊥
MC
. Prove that the points
X
,
D
,
Y
X,D,Y
X
,
D
,
Y
are collinear.
geometry