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Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2023 Spain Mathematical Olympiad
6
6
Part of
2023 Spain Mathematical Olympiad
Problems
(1)
Equal distance to triangle vertices
Source: Spain MO 2023 P6
3/12/2023
In an acute scalene triangle
A
B
C
ABC
A
BC
with incenter
I
I
I
, the line
A
I
AI
A
I
intersects the circumcircle again at
D
D
D
, and let
J
J
J
be a point such that
D
D
D
is the midpoint of
I
J
IJ
I
J
. Consider points
E
E
E
and
F
F
F
on line
B
C
BC
BC
such that
I
E
IE
I
E
and
J
F
JF
J
F
are perpendicular to
A
I
AI
A
I
. Consider points
G
G
G
on
A
E
AE
A
E
and
H
H
H
on
A
F
AF
A
F
such that
I
G
IG
I
G
and
J
H
JH
J
H
are perpendicular to
A
E
AE
A
E
and
A
F
AF
A
F
, respectively. Prove that
B
G
=
C
H
BG=CH
BG
=
C
H
.
geometry
Spain