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**.

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**.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestThis 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

Log in to reply

– Vishwash Kumar · 1 month ago

Eureka , Nice approach . But can be done even more simplyLog in to reply

Log in to reply

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

Log in to reply

– Neel Khare · 3 weeks, 1 day ago

this seems to be your signature dialogue hahahahaLog in to reply

– Rohit Camfar · 3 weeks ago

NoLog in to reply

– Rohit Camfar · 3 weeks ago

TRy trigLog in to reply

– Neel Khare · 3 weeks ago

It is it's a compliment not anything badLog in to reply

– Rohit Camfar · 3 weeks ago

I know thatLog in to reply

– Rohit Camfar · 3 weeks ago

Has your exam been postpondedLog in to reply

– Neel Khare · 3 weeks ago

No byeLog in to reply

– Rohit Camfar · 3 weeks ago

Exams are being held too lateLog in to reply

– Rohit Camfar · 3 weeks ago

Can;t believeLog in to reply