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Contests
International Contests
APMO
2019 APMO
3
3
Part of
2019 APMO
Problems
(1)
Variable point on the median
Source: APMO 2019 P3
6/11/2019
Let
A
B
C
ABC
A
BC
be a scalene triangle with circumcircle
Γ
\Gamma
Γ
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. A variable point
P
P
P
is selected in the line segment
A
M
AM
A
M
. The circumcircles of triangles
B
P
M
BPM
BPM
and
C
P
M
CPM
CPM
intersect
Γ
\Gamma
Γ
again at points
D
D
D
and
E
E
E
, respectively. The lines
D
P
DP
D
P
and
E
P
EP
EP
intersect (a second time) the circumcircles to triangles
C
P
M
CPM
CPM
and
B
P
M
BPM
BPM
at
X
X
X
and
Y
Y
Y
, respectively. Prove that as
P
P
P
varies, the circumcircle of
△
A
X
Y
\triangle AXY
△
A
X
Y
passes through a fixed point
T
T
T
distinct from
A
A
A
.
geometry
APMO