Assertion (A): If the centroid and circumcent | vedclass

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(A) The orthocenter $(H)$,centroid $(G)$,and circumcenter $(O)$ of a triangle are collinear,and the centroid $G$ divides the line segment joining the

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$A$ and $R$ are both independently true and $R$ is the correct explanation for $A$.

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(A) The orthocenter $(H)$,centroid $(G)$,and circumcenter $(O)$ of a triangle are collinear,and the centroid $G$ divides the line segment joining the orthocenter $H$ and the circumcenter $O$ in the ratio $2:1$.Let the orthocenter be $H(x, y)$,the centroid be $G(\alpha, \beta)$,and the circumcenter be $O(\gamma, \delta)$.By the section formula,$\alpha = \frac{x + 2\gamma}{3}$ and $\beta = \frac{y + 2\delta}{3}$.Solving for $x$ and $y$,we get $x = 3\alpha - 2\gamma$ and $y = 3\beta - 2\delta$.Thus,if the centroid and circumcenter are known,the orthocenter can be uniquely determined.Therefore,both $A$ and $R$ are true,and $R$ is the correct explanation for $A$.

Assertion $(A)$: If the centroid and circumcenter of a triangle are known,its orthocenter can be found.
Reason $(R)$: The centroid,orthocenter,and circumcenter of a triangle are collinear.

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Assertion $(A)$: If the centroid and circumcenter of a triangle are known,its orthocenter can be found.Reason $(R)$: The centroid,orthocenter,and circumcenter of a triangle are collinear.