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National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2014 NZMOC Camp Selection Problems
5
5
Part of
2014 NZMOC Camp Selection Problems
Problems
(1)
concyclic wanted, altitudes and midpoints related
Source: 2014 NZOMC Camp Selections p5
1/10/2021
Let
A
B
C
ABC
A
BC
be an acute angled triangle. Let the altitude from
C
C
C
to
A
B
AB
A
B
meet
A
B
AB
A
B
at
C
′
C'
C
′
and have midpoint
M
M
M
, and let the altitude from
B
B
B
to
A
C
AC
A
C
meet
A
C
AC
A
C
at
B
′
B'
B
′
and have midpoint
N
N
N
. Let
P
P
P
be the point of intersection of
A
M
AM
A
M
and
B
B
′
BB'
B
B
′
and
Q
Q
Q
the point of intersection of
A
N
AN
A
N
and
C
C
′
CC'
C
C
′
. Prove that the point
M
,
N
,
P
M, N, P
M
,
N
,
P
and
Q
Q
Q
lie on a circle.
geometry
Concyclic