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National and Regional Contests
Poland Contests
Poland - Second Round
2019 Poland - Second Round
6
6
Part of
2019 Poland - Second Round
Problems
(1)
Point in the interior
Source: 2019 Second Round - Poland
7/8/2019
Let
X
X
X
be a point lying in the interior of the acute triangle
A
B
C
ABC
A
BC
such that \begin{align*} \sphericalangle BAX = 2\sphericalangle XBA \ \ \ \ \hbox{and} \ \ \ \ \sphericalangle XAC = 2\sphericalangle ACX. \end{align*} Denote by
M
M
M
the midpoint of the arc
B
C
BC
BC
of the circumcircle
(
A
B
C
)
(ABC)
(
A
BC
)
containing
A
A
A
. Prove that
X
M
=
X
A
XM=XA
XM
=
X
A
.
geometry
lengths
angles