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SL/KL =? when AB + AC= 3BC, AC = 4AT, KL // BC, KL tangent to incircle of ABC

Source: 2021 Dutch BxMO TST p5

December 28, 2021

geometryratio

Problem Statement

Given is a triangle

ABC

with the property that

|AB| + |AC| = 3|BC|

. Let

T

be the point on segment

AC

such that

|AC| = 4|AT|

. Let

K

and

L

be points on the interior of line segments

AB

and

AC

respectively such that

KL \parallel BC

and

KL

is tangent to the inscribed circle of

\vartriangle ABC

. Let

S

be the intersection of

BT

and

KL

. Determine the ratio

\frac{|SL|}{|KL|}

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