Cevians AA', BB', and CC' are concurrent at the incenter I of ΔABC.

What is the value of

    |AI|      |BI|      |CI|
   ------- + ------- + -------
    |AA'|     |BB'|     |CC'|

in terms of the side lengths a, b, and c of ΔABC?

Can you prove it?

First thought:  the ratios should not be dependent on the side lengths because the units are different.  So if the the sum of the ratios can be given in terms of a, b, c it is probably a constant.

So I constructed the figure using geometers sketchpad and indeed the sum appears to be a constant 2.

Posted by Jer on 2011-08-25 13:58:29