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"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see here and here on MathWorld).

Consider the triangle formed by the centers of these circles, one can draw a new set of smaller Malfatti circles in this triangle. What is the limiting point of this process?

One thing sort of discouraging is that I tried on an isosceles triangle, unfortunately did not find the limiting point matching any of the known relevant points (e.g., incenter or the first Ajima-Malfatti point).

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I don't know the answer to your question, but it should be easy enough to compute this limit point numerically for an arbitrary triangle and use the result to search the Encyclopedia of Triangle Centers.

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  • $\begingroup$ @David, Thank you for your prompt reply! The website is extremely nice, but unfortunately, my numerical calculation (please feel free to cross-check) shows the first normalized trilinear coordinate is 1.4377 for a triangle with (a,b,c)=(6,9,13), which does not match any of the entries listed there. It's a little disappointing, but I will mark your answer as the accepted answer. $\endgroup$
    – Wiley
    Commented Jan 31, 2010 at 1:06
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    $\begingroup$ Disappointing? This means you can claim credit for a new center! $\endgroup$
    – David Eppstein
    Commented Jan 31, 2010 at 2:15

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