Proving Medians Split Each Other into a 1:2 ratio at their intersection point?

Or that two medians will break each other into 1/3 and 2/3 parts divided at their point of intersection.





How to prove this?Proving Medians Split Each Other into a 1:2 ratio at their intersection point?
Draw triangle ABC, draw medians AX, BY,%26amp; CZ


THe line joining midpoints of 2 sides of a triangle is parallel to 3rd side and half the 3rd side.


so ZY = 1/2 BC


Let G denote the point of intersection of three medians.


Let R denote the mid point of BG


and S denote the midpoint of GC.


In triangle GBC, R and S are midpoints of sides


so RS || BC and RS = 1/2 BC


and RS || ZY


RS = ZY = 1/2 BC


therefore ZYSR is paralleogram.


therefore ZG = GS diagonal of ||eogram bisect each other.


so GS = SC


But ZC + GS + SC = ZC (median)


3SC = ZC


SC = 1/3 ZC


ZS = 2/3 ZC


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