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National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2021 Dutch BxMO TST
5
5
Part of
2021 Dutch BxMO TST
Problems
(1)
SL/KL =? when AB + AC= 3BC, AC = 4AT, KL // BC, KL tangent to incircle of ABC
Source: 2021 Dutch BxMO TST p5
12/28/2021
Given is a triangle
A
B
C
ABC
A
BC
with the property that
∣
A
B
∣
+
∣
A
C
∣
=
3
∣
B
C
∣
|AB| + |AC| = 3|BC|
∣
A
B
∣
+
∣
A
C
∣
=
3∣
BC
∣
. Let
T
T
T
be the point on segment
A
C
AC
A
C
such that
∣
A
C
∣
=
4
∣
A
T
∣
|AC| = 4|AT|
∣
A
C
∣
=
4∣
A
T
∣
. Let
K
K
K
and
L
L
L
be points on the interior of line segments
A
B
AB
A
B
and
A
C
AC
A
C
respectively such that
K
L
∥
B
C
KL \parallel BC
K
L
∥
BC
and
K
L
KL
K
L
is tangent to the inscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
S
S
S
be the intersection of
B
T
BT
BT
and
K
L
KL
K
L
. Determine the ratio
∣
S
L
∣
∣
K
L
∣
\frac{|SL|}{|KL|}
∣
K
L
∣
∣
S
L
∣
geometry
ratio