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Serbia National Math Olympiad
2024 Serbia National Math Olympiad
4
4
Part of
2024 Serbia National Math Olympiad
Problems
(1)
Incircle-excircle config geo
Source: Serbia 2024 MO Problem 4
4/4/2024
Let
A
B
C
ABC
A
BC
be a triangle with incenter and
A
A
A
-excenter
I
,
I
a
I, I_a
I
,
I
a
, whose incircle touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
. The line
E
F
EF
EF
meets
B
C
BC
BC
at
P
P
P
and
X
X
X
is the midpoint of
P
D
PD
P
D
. Show that
X
I
⊥
D
I
a
XI \perp DI_a
X
I
⊥
D
I
a
.
geometry