Given a triangle $ABC$ with incenter $I$ and given the distances $AI$, $BI$ and $CI$.
What are the lengths $AB$, $AC$ and $BC$ of the sides?
Given the distances from incenter to vertices of a triangle, what are its sides' lengths?
1
$\begingroup$
geometry
triangles
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0Can you clarify what exactly is given? – 2017-02-26
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0@Lelouch What isn't clear ? – 2017-02-26
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0'Given a triangle ABC'. What exactly is given? The points A,B,C or what? – 2017-02-26
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0If you want to have it that way only the location of the incenter is given and the distances to the vertexes – 2017-02-26
1 Answers
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They are given by the roots of a cubic. By Stewart's theorem the squared length of the $A$ angle bisector is given by $\frac{bc}{(b+c)^2}\left((b+c)^2-a^2\right)$, so by Van Obel's theorem $AI^2=\frac{bc}{a^2}\left((b+c)^2- a^2\right)$.
To find $a,b,c$ from $AI^2,BI^2,CI^2$ is, in general, impossible with straightedge and compass only.