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National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2015 Flanders Math Olympiad
2
2
Part of
2015 Flanders Math Olympiad
Problems
(1)
Simple geometry
Source: 2015 Flanders Mathematical Olympiad, Problem 2
4/27/2015
Consider two points
Y
Y
Y
and
X
X
X
in a plane and a variable point
P
P
P
which is not on
X
Y
XY
X
Y
. Let the parallel line to
Y
P
YP
Y
P
through
X
X
X
intersect the internal angle bisector of
∠
X
Y
P
\angle XYP
∠
X
Y
P
in
A
A
A
, and let the parallel line to
X
P
XP
XP
through
Y
Y
Y
intersect the internal angle bisector of
∠
Y
X
P
\angle YXP
∠
Y
XP
in
B
B
B
. Let
A
B
AB
A
B
intersect
X
P
XP
XP
and
Y
P
YP
Y
P
in
S
S
S
and
T
T
T
respectively. Show that the product
∣
X
S
∣
∗
∣
Y
T
∣
|XS|*|YT|
∣
XS
∣
∗
∣
Y
T
∣
does not depend on the position of
P
P
P
.
geometry