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International Contests
Czech-Polish-Slovak Match
1999 Czech and Slovak Match
2
2
Part of
1999 Czech and Slovak Match
Problems
(1)
circumcircle of triangle passes through segment's midpoint
Source: Czech and Slovak Match 1999 P2
10/1/2017
The altitudes through the vertices
A
,
B
,
C
A,B,C
A
,
B
,
C
of an acute-angled triangle
A
B
C
ABC
A
BC
meet the opposite sides at
D
,
E
,
F
,
D,E,F,
D
,
E
,
F
,
respectively. The line through
D
D
D
parallel to
E
F
EF
EF
meets the lines
A
C
AC
A
C
and
A
B
AB
A
B
at
Q
Q
Q
and
R
R
R
, respectively. The line
E
F
EF
EF
meets
B
C
BC
BC
at
P
P
P
. Prove that the circumcircle of the triangle
P
Q
R
PQR
PQR
passes through the midpoint of
B
C
BC
BC
.
Concyclic
geometry