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Middle European Mathematical Olympiad
2015 Middle European Mathematical Olympiad
5
5
Part of
2015 Middle European Mathematical Olympiad
Problems
(1)
XY passes through D for certain X, Y
Source: MEMO 2015, problem T-5
8/28/2015
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
>
A
C
AB>AC
A
B
>
A
C
. Prove that there exists a point
D
D
D
with the following property: whenever two distinct points
X
X
X
and
Y
Y
Y
lie in the interior of
A
B
C
ABC
A
BC
such that the points
B
B
B
,
C
C
C
,
X
X
X
, and
Y
Y
Y
lie on a circle and
∠
A
X
B
−
∠
A
C
B
=
∠
C
Y
A
−
∠
C
B
A
\angle AXB-\angle ACB=\angle CYA-\angle CBA
∠
A
XB
−
∠
A
CB
=
∠
C
Y
A
−
∠
CB
A
holds, the line
X
Y
XY
X
Y
passes through
D
D
D
.
geometry