All possible values of the length of a segment

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All possible values of the length of a segment

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Source: Paraguayan National Olympiad 2009, Level 3, Problem 5

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#1 h Aug 31, 2014, 1:39 AM • 2 Y

Y by Adventure10, Mango247

In a triangle $ABC$ , let $I$ be its incenter. The distance from $I$ to the segment $BC$ is $4 cm$ and the distance from that point to vertex $B$ is $12 cm$ . Let $D$ be a point in the plane region between segments $AB$ and $BC$ such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$ . Find all possible values of the length $BD$ .

This post has been edited 1 time. Last edited by Leicich, Mar 25, 2018, 4:35 PM
Reason: Original post missed some crutial problem conditions; I have readded them now (thanks achen29!)

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#2 h Mar 16, 2018, 2:12 PM • 2 Y

Y by Adventure10, Mango247

Please change problem.. The correct version is
En un triángulo ABC, I es el incentro (punto de intersección de las tres bisectrices).
La distancia de I al lado BC es 4 cm y la distancia de I al vértice B es 12 cm.
Dentro de la región angular correspondiente a ABC se elige un punto D, tal que D sea el centro de una circunferencia tangente a las rectas AB y BC y que pase por el incentro.
Determinar los valores posibles de la distancia del punto D al vértice B.
http://www.omapa.org/wp-content/uploads/2015/08/ex-omapa-2009-juv-rf.pdf

which roughly translates to
In a triangle $ABC$ , let $I$ be its incenter. The distance from $I$ to $BC$ is $4$ and the length of segment BI is 12. Let $D$ be a point in the plane region between $AB$ and $BC$ , such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$ . Find all possible values of the length $BD$ .

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