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Moldova Team Selection Test
2024 Moldova Team Selection Test
2
2
Part of
2024 Moldova Team Selection Test
Problems
(1)
Perpendiculars to AD and tangent circles
Source: Moldova TST 2024 P2
6/9/2024
In the acute-angled triangle
A
B
C
ABC
A
BC
, let
A
D
AD
A
D
,
D
∈
B
C
D \in BC
D
∈
BC
be the
A
A
A
-angle bisector. The perpenducular to
B
C
BC
BC
through
D
D
D
and the perpendicular to
A
D
AD
A
D
through
A
A
A
meet at
I
I
I
. The circle with center
I
I
I
and radius
I
D
ID
I
D
, intersects
A
B
AB
A
B
and
A
C
AC
A
C
at
F
F
F
and
E
E
E
respectively. On the arc
F
E
FE
FE
, which does not contain
A
A
A
, of the circumcircle of
A
F
E
AFE
A
FE
, consider a point
X
X
X
, such that
X
F
X
E
=
A
F
A
E
\frac{XF}{XE}=\frac{AF}{AE}
XE
XF
=
A
E
A
F
. Prove that the circumcircles of triangles
A
F
E
AFE
A
FE
and
B
X
C
BXC
BXC
are tangent.
geometry