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IMSC
IMSC 2023
4
4
Part of
IMSC 2023
Problems
(1)
Incircle geo with equal segments
Source: IMSC 2023 Mock IMO P4
7/13/2023
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
and let
A
I
AI
A
I
meet
B
C
BC
BC
at
D
D
D
. Let
E
E
E
be a point on the segment
A
C
AC
A
C
, such that
C
D
=
C
E
CD=CE
C
D
=
CE
and let
F
F
F
be on the segment
A
B
AB
A
B
such that
B
F
=
B
D
BF=BD
BF
=
B
D
. Let
(
C
E
I
)
∩
(
D
F
I
)
=
P
≠
I
(CEI) \cap (DFI)=P \neq I
(
CE
I
)
∩
(
D
F
I
)
=
P
=
I
and
(
B
F
I
)
∩
(
D
E
I
)
=
Q
≠
I
(BFI) \cap (DEI)=Q \neq I
(
BF
I
)
∩
(
D
E
I
)
=
Q
=
I
. Prove that
P
Q
⊥
B
C
PQ \perp BC
PQ
⊥
BC
.Proposed by Leonardo Franchi, Italy
geometry