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Contests
National and Regional Contests
New Zealand Contests
New Zealand MO
2020 New Zealand MO
6
6
Part of
2020 New Zealand MO
Problems
(1)
PY bisects AX, feet of altitudes, intersections of circumcircles AXC and PQC
Source: 2020 NZMO Round 1 p6
1/10/2021
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an acute triangle with
A
B
>
A
C
AB > AC
A
B
>
A
C
. Let
P
P
P
be the foot of the altitude from
C
C
C
to
A
B
AB
A
B
and let
Q
Q
Q
be the foot of the altitude from
B
B
B
to
A
C
AC
A
C
. Let
X
X
X
be the intersection of
P
Q
PQ
PQ
and
B
C
BC
BC
. Let the intersection of the circumcircles of triangle
△
A
X
C
\vartriangle AXC
△
A
XC
and triangle
△
P
Q
C
\vartriangle PQC
△
PQC
be distinct points:
C
C
C
and
Y
Y
Y
. Prove that
P
Y
PY
P
Y
bisects
A
X
AX
A
X
.
geometry
bisects segment