Incenter by vector algebra

Question

If !!ABC!! is a triangle, !!P!! is a point outside the triangle.

Then !!avecPA+bvecPB+cvecPC=(a+b+c)vecPK!! holds if K is?

  • Circumcenter
  • Centroid
  • Any point
  • None
  • Incenter

Solution

We will modify this equation and elminate P so we get !!avecPA+bvecPB+cvecPC=(a+b+c)vecPK!!

!!avecA+bvecB+cvecC=(a+b+c)vecK!!

!!vecK = (avecA+bvecB+cvecC)/(a+b+c)!!

Here it is better if we remember that !!K!! is an incenter.

Though it is recommended to prove that !!K!! is an incenter.

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