The locus of a point which moves such that th | vedclass

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(B) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.Let the moving point be $P(x, y)$.The sum of the squares of the

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Centroid of the triangle

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(B) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.Let the moving point be $P(x, y)$.The sum of the squares of the distances from $P$ to the vertices is constant,say $K$:$(x - x_1)^2 + (y - y_1)^2 + (x - x_2)^2 + (y - y_2)^2 + (x - x_3)^2 + (y - y_3)^2 = K$Expanding the terms:$3x^2 - 2x(x_1 + x_2 + x_3) + 3y^2 - 2y(y_1 + y_2 + y_3) + (x_1^2 + x_2^2 + x_3^2 + y_1^2 + y_2^2 + y_3^2) = K$Dividing by $3$:$x^2 - 2x\left(\frac{x_1 + x_2 + x_3}{3}ight) + y^2 - 2y\left(\frac{y_1 + y_2 + y_3}{3}ight) = 	ext{constant}$This is the equation of a circle with centre at $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}ight)$,which is the centroid of the triangle.

The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is constant,is a circle whose centre is at the

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