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National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2017 NZMOC Camp Selection Problems
6
6
Part of
2017 NZMOC Camp Selection Problems
Problems
(1)
concyclic wanted, 2 circumcircles related
Source: 2017 NZOMC Camp Selections p6
1/10/2021
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral. The circumcircle of the triangle
A
B
C
ABC
A
BC
intersects the sides
C
D
CD
C
D
and
D
A
DA
D
A
in the points
P
P
P
and
Q
Q
Q
respectively, while the circumcircle of
C
D
A
CDA
C
D
A
intersects the sides
A
B
AB
A
B
and
B
C
BC
BC
in the points
R
R
R
and
S
S
S
. The lines
B
P
BP
BP
and
B
Q
BQ
BQ
intersect the line
R
S
RS
RS
in the points
M
M
M
and
N
N
N
respectively. Prove that the points
M
,
N
,
P
M, N, P
M
,
N
,
P
and
Q
Q
Q
lie on the same circle.
geometry
Concyclic