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Regional Olympiad Problem

Prove that the centroid of a triangle divides the median in the ratio of \(2:1\).

Bonus: Do it without using co-ordinate geometry but by using geometry.

Note by Rohit Camfar
1 month ago

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This can be easily done.
Consider a triangle ABC with centroid O and medians AD; BE and CF.
Now;
Let; \(ar(AOE) = ar(COE) = a\);
\(ar(COD) = ar(BOD) = b\);
\(ar(BOF) = ar(AOF) = c\);

But; since \(ar(ADB) = ar(BEA) = \frac{ar(ABC)}{2} \)
\( \implies ar(AOE) = ar(BOD)\)
OR \(a = b\)

Similarly, we can prove that in fact;
\(a = b = c = \frac{ar(ABC)}{6} \)

Now, using the fact that triangles AOF and AOC share the same height but the area of the latter is twice the former; we can easily deduce that CO:FO = 2:1.

Similarly we can do for the other 2 medians.

Sorry for being unable to upload picture. Yatin Khanna · 1 month ago

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@Yatin Khanna Eureka , Nice approach . But can be done even more simply Vishwash Kumar · 1 month ago

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Vishwash Kumar · 1 month ago

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Keep it pure keep it geometric . Vishwash Kumar · 1 month ago

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@Vishwash Kumar this seems to be your signature dialogue hahahaha Neel Khare · 3 weeks, 1 day ago

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@Neel Khare No Rohit Camfar · 3 weeks ago

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@Rohit Camfar TRy trig Rohit Camfar · 3 weeks ago

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@Rohit Camfar It is it's a compliment not anything bad Neel Khare · 3 weeks ago

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@Neel Khare I know that Rohit Camfar · 3 weeks ago

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@Rohit Camfar Has your exam been postponded Rohit Camfar · 3 weeks ago

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@Rohit Camfar No bye Neel Khare · 3 weeks ago

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@Neel Khare Exams are being held too late Rohit Camfar · 3 weeks ago

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@Rohit Camfar Can;t believe Rohit Camfar · 3 weeks ago

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