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IMO Shortlist
2016 IMO Shortlist
G7
G7
Part of
2016 IMO Shortlist
Problems
(1)
Reflections of lines through reflections of excenters
Source: 2016 IMO Shortlist G7
7/19/2017
Let
I
I
I
be the incentre of a non-equilateral triangle
A
B
C
ABC
A
BC
,
I
A
I_A
I
A
be the
A
A
A
-excentre,
I
A
′
I'_A
I
A
′
be the reflection of
I
A
I_A
I
A
in
B
C
BC
BC
, and
l
A
l_A
l
A
be the reflection of line
A
I
A
′
AI'_A
A
I
A
′
in
A
I
AI
A
I
. Define points
I
B
I_B
I
B
,
I
B
′
I'_B
I
B
′
and line
l
B
l_B
l
B
analogously. Let
P
P
P
be the intersection point of
l
A
l_A
l
A
and
l
B
l_B
l
B
.[*] Prove that
P
P
P
lies on line
O
I
OI
O
I
where
O
O
O
is the circumcentre of triangle
A
B
C
ABC
A
BC
. [*] Let one of the tangents from
P
P
P
to the incircle of triangle
A
B
C
ABC
A
BC
meet the circumcircle at points
X
X
X
and
Y
Y
Y
. Show that
∠
X
I
Y
=
12
0
∘
\angle XIY = 120^{\circ}
∠
X
I
Y
=
12
0
∘
.
geometry
IMO Shortlist