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2023 USAMO
6
6
Part of
2023 USAMO
Problems
(1)
force overlay inversion vibes
Source: USAMO 2023/6
3/23/2023
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
and excenters
I
a
I_a
I
a
,
I
b
I_b
I
b
, and
I
c
I_c
I
c
opposite
A
A
A
,
B
B
B
, and
C
C
C
, respectively. Let
D
D
D
be an arbitrary point on the circumcircle of
△
A
B
C
\triangle{ABC}
△
A
BC
that does not lie on any of the lines
I
I
a
II_a
I
I
a
,
I
b
I
c
I_bI_c
I
b
I
c
, or
B
C
BC
BC
. Suppose the circumcircles of
△
D
I
I
a
\triangle{DII_a}
△
D
I
I
a
and
△
D
I
b
I
c
\triangle{DI_bI_c}
△
D
I
b
I
c
intersect at two distinct points
D
D
D
and
F
F
F
. If
E
E
E
is the intersection of lines
D
F
DF
D
F
and
B
C
BC
BC
, prove that
∠
B
A
D
=
∠
E
A
C
\angle{BAD} = \angle{EAC}
∠
B
A
D
=
∠
E
A
C
.Proposed by Zach Chroman
USAMO
geometry
Hi