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Contests
National and Regional Contests
Switzerland Contests
Swiss NMO - geometry
2022.8
2022.8
Part of
Swiss NMO - geometry
Problems
(1)
AX is independent of the choice of P, XI_1 _|_ XI_2, 2 incenters
Source: Switzerland - 2022 Swiss Final Round p8
11/17/2022
Let
A
B
C
ABC
A
BC
be a triangle and let
P
P
P
be a point in the interior of the side
B
C
BC
BC
. Let
I
1
I_1
I
1
and
I
2
I_2
I
2
be the incenters of the triangles
A
P
B
AP B
A
PB
and
A
P
C
AP C
A
PC
, respectively. Let
X
X
X
be the closest point to
A
A
A
on the line
A
P
AP
A
P
such that
X
I
1
XI_1
X
I
1
is perpendicular to
X
I
2
XI_2
X
I
2
. Prove that the distance
A
X
AX
A
X
is independent of the choice of
P
P
P
.
geometry
incenter
fixed