MathDB
Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2023 Philippine MO
3
3
Part of
2023 Philippine MO
Problems
(1)
Circumcircles and concurrence
Source: Philippine MO 2023/3
3/19/2023
In
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
>
A
C
AB > AC
A
B
>
A
C
. Point
P
P
P
is on line
B
C
BC
BC
such that
A
P
AP
A
P
is tangent to its circumcircle. Let
M
M
M
be the midpoint of
A
B
AB
A
B
, and suppose the circumcircle of
△
P
M
A
\triangle PMA
△
PM
A
meets line
A
C
AC
A
C
again at
N
N
N
. Point
Q
Q
Q
is the reflection of
P
P
P
with respect to the midpoint of segment
B
C
BC
BC
. The line through
B
B
B
parallel to
Q
N
QN
QN
meets
P
N
PN
PN
at
D
D
D
, and the line through
P
P
P
parallel to
D
M
DM
D
M
meets the circumcircle of
△
P
M
B
\triangle PMB
△
PMB
again at
E
E
E
. Show that the lines
P
M
PM
PM
,
B
E
BE
BE
, and
A
C
AC
A
C
are concurrent.
geometry
circumcircle
PMO