# medians in a triangle

Is it true that the fact that medians of any two sides meet at a distance of 2/3 from vertex to median on the opposite side implies that all three medians intersect at a single point?

If yes, how is it so?

• – lab bhattacharjee Nov 25 '13 at 15:36
• Yes, but if you want to prove this fact can you make use of the fact that...and how? I suppose that is the question. I hope it makes sense. wink – Adam Nov 25 '13 at 15:41

• Right. Two lines intersect at most once. This means that each median intersects another two-thirds of the way from the vertex to edge. Take the median from $A$ and let $X$ be the point two-thirds of the way along the median. Then the median from $B$ meets the median from $A$ at $X$ and the median from $C$ meets the median from $A$ at $X$. Hence all three meet at $X$. – EuYu Nov 25 '13 at 16:15