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Kvant 2021
M2672
M2672
Part of
Kvant 2021
Problems
(1)
Incircle geometry
Source: Kvant Magazine No. 10 2021 M2672
3/9/2023
Let the inscribed circle
ω
\omega
ω
of the triangle
A
B
C
ABC
A
BC
have a center
I
I{}
I
and touch the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
at points
D
,
E
D, E
D
,
E
and
F
F{}
F
respectively. Let
M
M{}
M
and
N
N{}
N
be points on the straight line
E
F
EF
EF
such that
B
M
∥
A
C
BM \parallel AC
BM
∥
A
C
and
C
N
∥
A
B
CN \parallel AB
CN
∥
A
B
. Let
P
P{}
P
and
Q
Q{}
Q
be points on the segments
D
M
DM{}
D
M
and
D
N
DN{}
D
N
, respectively, such that
B
P
∥
C
Q
BP \parallel CQ
BP
∥
CQ
. Prove that the intersection point of the lines
P
F
PF
PF
and
Q
E
QE
QE
lies on
ω
\omega
ω
.Proposed by Don Luu (Vietnam)
geometry
Kvant