Index Of Triangle 2009 New May 2026

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In some geometry problems (especially from Asian contests), the index of a triangle is defined for a point ( P ) inside triangle ( ABC ) as: Do not just type the phrase into Google

[ \textIndex(P) = \frac[PBC][ABC] + \frac[PCA][ABC] + \frac[PAB][ABC] ] Let ( P ) be a point inside triangle ( ABC )

But that sum always equals 1. So that's trivial.
A more interesting definition (seen in some 2009 problems) is:

Let ( P ) be a point inside triangle ( ABC ). Define the index of triangle ( ABC ) with respect to ( P ) as: [ I(P) = \fracPAr_a + \fracPBr_b + \fracPCr_c ] where ( r_a, r_b, r_c ) are distances from ( P ) to sides ( BC, CA, AB ) respectively.

But that’s unusual.