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

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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.
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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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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 :
 

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