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National and Regional Contests
China Contests
XES Mathematics Olympiad
the 7th XMO
the 7th XMO
Part of
XES Mathematics Olympiad
Subcontests
(1)
1
1
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O_1O_2_|_CF , line connecting circumcenters, starting with isosceles
As shown in the figure, it is known that
B
C
=
A
C
BC = AC
BC
=
A
C
in
A
B
C
ABC
A
BC
,
M
M
M
is the midpoint of
A
B
AB
A
B
, points
D
D
D
and
E
E
E
lie on
A
B
AB
A
B
satisfying
∠
D
C
E
=
∠
M
C
B
\angle DCE = \angle MCB
∠
D
CE
=
∠
MCB
, the circumscribed circle of
△
B
D
C
\vartriangle BDC
△
B
D
C
and the circumscribed circle of
△
A
E
C
\vartriangle AEC
△
A
EC
intersect at point
F
F
F
(different from point
C
C
C
), point
H
H
H
lies on
A
B
AB
A
B
such that the straight line
C
M
CM
CM
bisects the line segment
H
F
HF
H
F
. Let the circumcenters of
△
H
F
E
\vartriangle HFE
△
H
FE
and
△
B
F
M
\vartriangle BFM
△
BFM
be
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively. Prove that
O
1
O
2
⊥
C
F
O_1O_2\perp CF
O
1
O
2
⊥
CF
. https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png