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Spain Contests
Spain Mathematical Olympiad
1985 Spain Mathematical Olympiad
6
6
Part of
1985 Spain Mathematical Olympiad
Problems
(1)
intersecting rays, antiparallels, angle bisectors, find point
Source: Spanish Mathematical Olympiad 1985 P6
8/2/2018
Let
O
X
OX
OX
and
O
Y
OY
O
Y
be non-collinear rays. Through a point
A
A
A
on
O
X
OX
OX
, draw two lines
r
1
r_1
r
1
and
r
2
r_2
r
2
that are antiparallel with respect to
∠
X
O
Y
\angle XOY
∠
XO
Y
. Let
r
1
r_1
r
1
cut
O
Y
OY
O
Y
at
M
M
M
and
r
2
r_2
r
2
cut
O
Y
OY
O
Y
at
N
N
N
. (Thus,
∠
O
A
M
=
∠
O
N
A
\angle OAM = \angle ONA
∠
O
A
M
=
∠
ON
A
). The bisectors of
∠
A
M
Y
\angle AMY
∠
A
M
Y
and
∠
A
N
Y
\angle ANY
∠
A
N
Y
meet at
P
P
P
. Determine the location of
P
P
P
.
geometry
angle bisector