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National and Regional Contests
Canada Contests
Canada National Olympiad
2016 Canada National Olympiad
5
5
Part of
2016 Canada National Olympiad
Problems
(1)
Concurrency on Circumcircle
Source: 2016 CMO #5
4/10/2016
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute-angled triangle with altitudes
A
D
AD
A
D
and
B
E
BE
BE
meeting at
H
H
H
. Let
M
M
M
be the midpoint of segment
A
B
AB
A
B
, and suppose that the circumcircles of
△
D
E
M
\triangle DEM
△
D
EM
and
△
A
B
H
\triangle ABH
△
A
B
H
meet at points
P
P
P
and
Q
Q
Q
with
P
P
P
on the same side of
C
H
CH
C
H
as
A
A
A
. Prove that the lines
E
D
,
P
H
,
ED, PH,
E
D
,
P
H
,
and
M
Q
MQ
MQ
all pass through a single point on the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
.
geometry
circumcircle