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Middle European Mathematical Olympiad
2022 Middle European Mathematical Olympiad
5
5
Part of
2022 Middle European Mathematical Olympiad
Problems
(1)
Right triangle geo with tangents
Source: MEMO 2022 T5
9/2/2022
Let
Ω
\Omega
Ω
be the circumcircle of a triangle
A
B
C
ABC
A
BC
with
∠
C
A
B
=
90
\angle CAB = 90
∠
C
A
B
=
90
. The medians through
B
B
B
and
C
C
C
meet
Ω
\Omega
Ω
again at
D
D
D
and
E
E
E
, respectively. The tangent to
Ω
\Omega
Ω
at
D
D
D
intersects the line
A
C
AC
A
C
at
X
X
X
and the tangent to
Ω
\Omega
Ω
at
E
E
E
intersects the line
A
B
AB
A
B
at
Y
Y
Y
. Prove that the line
X
Y
XY
X
Y
is tangent to
Ω
\Omega
Ω
.
geometry