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(D) Let the orthocenter be $H(4, -1)$ and the centroid be $G(2, 1)$.Let the circumcenter be $O(x, y)$.We know that the centroid $G$ divides the line s

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None of these

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(D) Let the orthocenter be $H(4, -1)$ and the centroid be $G(2, 1)$.Let the circumcenter be $O(x, y)$.We know that the centroid $G$ divides the line segment joining the orthocenter $H$ and the circumcenter $O$ in the ratio $2:1$.Using the section formula,$G = \left( \frac{1 \cdot x_H + 2 \cdot x_O}{1+2}, \frac{1 \cdot y_H + 2 \cdot y_O}{1+2} \right)$.Substituting the values: $(2, 1) = \left( \frac{4 + 2x}{3}, \frac{-1 + 2y}{3} \right)$.Equating the $x$-coordinates: $2 = \frac{4 + 2x}{3} \implies 6 = 4 + 2x \implies 2x = 2 \implies x = 1$.Equating the $y$-coordinates: $1 = \frac{-1 + 2y}{3} \implies 3 = -1 + 2y \implies 2y = 4 \implies y = 2$.Thus,the coordinates of the circumcenter are $(1, 2)$.Since $(1, 2)$ is not among the given options,the correct answer is $D$.

If the coordinates of the orthocenter and the centroid of a triangle are $(4, -1)$ and $(2, 1)$ respectively,then what are the coordinates of the circumcenter of the triangle?

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