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National and Regional Contests
China Contests
XES Mathematics Olympiad
XMO (China) 2-15 - geometry
XMO (China) 2-15 - geometry
Part of
XES Mathematics Olympiad
Subcontests
(1)
15.1
1
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tangent circles wanted, 2 right angles, 4 midpoints related
As shown in the figure, in the quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
B
⊥
B
C
AB\perp BC
A
B
⊥
BC
,
A
D
⊥
C
D
AD\perp CD
A
D
⊥
C
D
, let
E
E
E
be a point on line
B
D
BD
B
D
such that
E
C
=
C
A
EC = CA
EC
=
C
A
. The line perpendicular on line
A
C
AC
A
C
passing through
E
E
E
, intersects line
A
B
AB
A
B
at point
F
F
F
, and line
A
D
AD
A
D
at point
G
G
G
. Let
X
X
X
and
Y
Y
Y
the midpoints of line segments
A
F
AF
A
F
and
A
G
AG
A
G
respectively. Let
Z
Z
Z
and
W
W
W
be the midpoints of line segments
B
E
BE
BE
and
D
E
DE
D
E
respectively. Prove that the circumscribed circle of
△
W
B
X
\vartriangle WBX
△
W
BX
is tangent to the circumscribed circle of
△
Z
D
Y
\vartriangle ZDY
△
Z
D
Y
. https://cdn.artofproblemsolving.com/attachments/0/3/1f6fca7509e6fd6cad662b42abd236fd4858ca.jpg