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Kvant Problems
Kvant 2023
M2764
M2764
Part of
Kvant 2023
Problems
(1)
Concurrency geometry
Source: Kvant Magazine No. 9 2023 M2764
1/9/2024
Let
B
E
BE{}
BE
and
C
F
CF
CF
be heights in the acute-angled triangle
A
B
C
ABC{}
A
BC
and let
O
O{}
O
be its circumcenter. The points
M
M{}
M
and
N
N{}
N
are selected on the side
B
C
BC{}
BC
so that
B
M
=
C
N
.
BM=CN.{}
BM
=
CN
.
The line
B
E
BE{}
BE
intersects the circle
(
M
B
F
)
(MBF)
(
MBF
)
a second time at
P
P{}
P
and the line
C
F
CF{}
CF
intersects the circle
(
N
C
E
)
(NCE)
(
NCE
)
a second time at
Q
.
Q.{}
Q
.
Prove that the lines
P
F
,
Q
E
PF, QE
PF
,
QE
and
A
O
AO{}
A
O
intersect at the same point.Proposed by Luu Dong
geometry
concurrency