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National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2022 Macedonian Team Selection Test
Problem 4
Problem 4
Part of
2022 Macedonian Team Selection Test
Problems
(1)
Circles through A, tangent to DE
Source: Macedonian TST 2022, P4
5/21/2022
Given is an acute triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB<AC
A
B
<
A
C
with altitudes
B
D
BD
B
D
and
C
E
CE
CE
. Let the tangents to the circumcircle at
B
B
B
and
C
C
C
meet at
Y
Y
Y
. Let
ω
1
\omega_1
ω
1
be the circle through
A
A
A
tangent to
D
E
DE
D
E
at
E
E
E
; define
ω
2
\omega_2
ω
2
similarly, and let their intersection point be
X
X
X
. Prove that
A
,
X
,
Y
A, X, Y
A
,
X
,
Y
are colinear.
<
s
p
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a
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=
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−
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t
a
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′
>
P
r
o
p
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s
e
d
b
y
N
i
k
o
l
a
V
e
l
o
v
<
/
s
p
a
n
>
<span class='latex-italic'>Proposed by Nikola Velov</span>
<
s
p
an
c
l
a
ss
=
′
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a
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x
−
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t
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c
′
>
P
ro
p
ose
d
b
y
N
ik
o
l
aV
e
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<
/
s
p
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>
geometry