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inradius http://sphotos-a.ak.fbcdn.net/hphotos-ak-ash4/603253_4699189150138_1902686882_n.jpg

Inside triangle ABC there are three circles with radius $r_1$, $r_2$, and $r_3$ each of which is tangent to two sides of the triangle and to its incircle with radius r. All of $r$, $r_1$, $r_2$, and $r_3$ are distinct perfect square integers. Find the smallest value of inradius $r$.

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    Nice problem. Where is it from?2012-12-27
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    Saw it on a math group. :)2012-12-27
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    Is there a requirement that the circle with radius $r$ is tangent to all three sides of the triangle?2012-12-27
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    It is the incircle so it must be tangent to all three sides.2012-12-27
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    A hint for a solution method (I could try it, but can't right now): Notice that each small circle is homothetic to the incircle so you should be able to find the ratios $r/r_i$ in terms of $a,b,c$. After that I don't know what you could do. I guess you may need some assumptions on $a,b,c$, also.2012-12-27
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    What does homothetic mean?2012-12-27
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    The little circles are extraneous. Given an angle $A$, $B$, or $C$ the radius of the incircle must be on the line that bisects the angle. Fortunately the three angle bisectors meet at a common point. This point is the center of the incircle. It has been a long time since I last chased around a triangle to find a length. Perhaps someone else can turn this observation into an answer.2012-12-28
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    See my comment on the incomplete answer below, which I believe finishes the problem.2012-12-28

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