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National and Regional Contests
Turkey Contests
Turkey EGMO TST
2014 Turkey EGMO TST
1
1
Part of
2014 Turkey EGMO TST
Problems
(1)
Concyclicity Implies Perpendicularity
Source: Turkey EGMO TST 2014 P1
8/6/2016
Let
D
D
D
be the midpoint of the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
and
A
D
AD
A
D
intersect the circumcircle of
A
B
C
ABC
A
BC
for the second time at
E
E
E
. Let
P
P
P
be the point symmetric to the point
E
E
E
with respect to the point
D
D
D
and
Q
Q
Q
be the point of intersection of the lines
C
P
CP
CP
and
A
B
AB
A
B
. Prove that if
A
,
C
,
D
,
Q
A,C,D,Q
A
,
C
,
D
,
Q
are concyclic, then the lines
B
P
BP
BP
and
A
C
AC
A
C
are perpendicular.
geometry
circumcircle