InterviewSolution
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InterviewSolution
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lf G be the centroid of a triangle ABC and P be any other point in the plane prove that `PA^2+PB^2+PC^2=GA^2+GB^2+GC^2+3GP^2` |
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Answer» `G ` is intersection of median `D` is mid[point of DC `G(0,0)` Coordinates of D=`((c + e)/2, (d+f)/2)` centroid of triangle G=`((a+c+e)/3 , (b+d+f)/3)` `(a+c+e)/3 = 0& (b+d+f)/3 = 0` `e= -(a+c) & f= - (b+d)` `PA^2 + PB^2 + PC^2 = GA^2 + GB^2 + GC^3 + 3GP^2` `[(x-a)^2 + (y-b)^2] + [(x-c)^2+ (y-d)^2] + [(x+a+c)^2 + (y+b+d)^2]` `= [(0-a)^2 + (0-b)^2] + [(0-c)^2 + (0-d)^2] + [(a+c)^2 + (b+d)^2] + 3[(x-0)^2 + (y-0)^2]` from LHS `x^2 + a^2 - 2ax + y^2 + b^2 - 2by + x^2 + c^2 - 2cx + y^2 + d^2 = 2dy + x^2 + a^2 + c^2 ` `= 3(x^2 + y^2) + 2(a^2 + b^2 +c^2 + d^2+ c^2 + d^2 + ca+bd)` from RHS `a^2 + b^2 + c^2 + d^2 + a^2 + c^2 + 2ac + b^2 + d^2 + 2bd + 3(x^2 + y^2)` `= 3[x^2 + y^2] + 2(a^2 + b^2 + c^2 + d^2 + ac+bd)` LHS=RHS Hence proved |
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