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Balkan MO
2019 Balkan MO
3
3
Part of
2019 Balkan MO
Problems
(1)
Concurrence, Isogonality
Source: BMO 2019, Problem 3
5/2/2019
Let
A
B
C
ABC
A
BC
be an acute scalene triangle. Let
X
X
X
and
Y
Y
Y
be two distinct interior points of the segment
B
C
BC
BC
such that
∠
C
A
X
=
∠
Y
A
B
\angle{CAX} = \angle{YAB}
∠
C
A
X
=
∠
Y
A
B
. Suppose that:
1
)
1)
1
)
K
K
K
and
S
S
S
are the feet of the perpendiculars from from
B
B
B
to the lines
A
X
AX
A
X
and
A
Y
AY
A
Y
respectively.
2
)
2)
2
)
T
T
T
and
L
L
L
are the feet of the perpendiculars from
C
C
C
to the lines
A
X
AX
A
X
and
A
Y
AY
A
Y
respectively. Prove that
K
L
KL
K
L
and
S
T
ST
ST
intersect on the line
B
C
BC
BC
.
geometry
BMO