MHB How Does the Incenter Position Relate to Triangle Side Lengths?

Thread starter Albert1
Start date Sep 14, 2014

AI Thread Summary

The incenter $ I $ of triangle $ ABC $ is defined in relation to the triangle's sides $ a $, $ b $, and $ c $ opposite vertices $ A $, $ B $, and $ C $ respectively. The distances from the incenter to the vertices are denoted as $ IA = x $, $ IB = y $, and $ IC = z $. The key proof to establish is that $ ax^2 + by^2 + cz^2 = abc $. This relationship highlights how the lengths from the incenter to the vertices are influenced by the triangle's side lengths. Understanding this connection is crucial for deeper insights into triangle geometry.

Sep 14, 2014

Albert1

Point $I$ is the incenter of $\triangle ABC $
giving :$BC=a , \, AC=b,\, AB=c$
$IA=x, \, IB=y, \, IC=z$
prove :$ax^2+by^2+cz^2=abc$

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Sep 23, 2014

Albert1

Albert said:

Point $I$ is the incenter of $\triangle ABC $
giving :$BC=a , \, AC=b,\, AB=c$
$IA=x, \, IB=y, \, IC=z$
prove :$ax^2+by^2+cz^2=abc$

soluton :

Last edited: Sep 23, 2014

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