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National and Regional Contests
Canada Contests
Canada National Olympiad
2013 Canada National Olympiad
3
3
Part of
2013 Canada National Olympiad
Problems
(1)
Intersections of Circumcircles and Hypotenuse
Source: 2013 CMO #3
3/31/2013
Let
G
G
G
be the centroid of a right-angled triangle
A
B
C
ABC
A
BC
with
∠
B
C
A
=
9
0
∘
\angle BCA = 90^\circ
∠
BC
A
=
9
0
∘
. Let
P
P
P
be the point on ray
A
G
AG
A
G
such that
∠
C
P
A
=
∠
C
A
B
\angle CPA = \angle CAB
∠
CP
A
=
∠
C
A
B
, and let
Q
Q
Q
be the point on ray
B
G
BG
BG
such that
∠
C
Q
B
=
∠
A
B
C
\angle CQB = \angle ABC
∠
CQB
=
∠
A
BC
. Prove that the circumcircles of triangles
A
Q
G
AQG
A
QG
and
B
P
G
BPG
BPG
meet at a point on side
A
B
AB
A
B
.
geometry
circumcircle