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The orthocentre of a triangle is $(-3,5)$ and circumcentre is $(6,2)$ then find the centroid of the triangle.

My Attempt:

We know that the centroid is the point of intersection of the three medians of a triangle. It divides the median in the ratio of $2:1$.

Now, how do I proceed further?

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Hint Eulers line tells us that the centroid divides the line joining circumcenter and orthocenter in the ratio $2:1$ where $\frac {cg}{go}=\frac {2}{1} $

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  • $\begingroup$ I couldn't understand. Please elaborate. $\endgroup$ – pi-π Apr 16 '17 at 13:53
  • $\begingroup$ If circumcenter=c,g=centroid,o=orthocenter then distance between cg=2go can you now find point using internal dvision formula. Hope you got it $\endgroup$ – Archis Welankar Apr 16 '17 at 13:54
  • $\begingroup$ Euler's line says that the centroid is 2/3rd along the line joining the circumcentre and orthocentre. $\endgroup$ – Toby Mak Apr 16 '17 at 13:54
  • $\begingroup$ @ArchisWelankar, You mean $$(x,y)=(\dfrac {m_1x_2+m_2x_1}{m_1+m_2} , \dfrac {m_1y_2+m_2y_1}{m_1+m_2})$$?????? $\endgroup$ – pi-π Apr 16 '17 at 15:18
  • $\begingroup$ Yes exactly. ..... $\endgroup$ – Archis Welankar Apr 16 '17 at 16:19

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