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Final Mathematical Cup
2019 Final Mathematical Cup
1
1
Part of
2019 Final Mathematical Cup
Problems
(1)
AP bisects BC, BD // CE, 3 circumcircles related
Source: 1st Final Mathematical Cup 2019 FMC , juniors p1
10/6/2020
Let
A
B
C
ABC
A
BC
be a triangle and let
D
,
E
D, E
D
,
E
are points on its circumscribed circle, such that
D
D
D
lies on arc
A
B
,
E
AB, E
A
B
,
E
lies on arc
A
C
AC
A
C
(smaller arcs) and
B
D
∥
C
E
BD \parallel CE
B
D
∥
CE
. Let the point F be the intersection of the lines
D
A
DA
D
A
and
C
E
CE
CE
, and the intersection of the lines
E
A
EA
E
A
and
B
D
BD
B
D
is
G
G
G
. Let
P
P
P
be the second intersection of the circumscribed circles of
△
A
B
G
\vartriangle ABG
△
A
BG
and
△
A
C
F
\vartriangle ACF
△
A
CF
. Prove that the line
A
P
AP
A
P
passes through the mid point of the side
B
C
BC
BC
.
geometry
circumcircle