Question

Where can the medians of a triangle intersect?

I. inside the triangle
II. on the triangle
III. outside the triangle

Answer 1

...

Answer 2

Let's begin with a drawing.
We take any triangle ABC and draw two of the medians.
By definition the "median" of a triangle is the line that goes from one of the vertices to the middle point of the opposite side.
So, let's take a triangle and draw two of them:

What I want to prove is that the third median will go through that "G" point.

Answer 3

so it's only inside the triangle correct ?

Answer 4

on the triangle

Answer 5

I'll begin by Saying that triangle BMN and ABC are similar (I'll leave that proof to you).
And by that we can deduce that MN=AC/2 and with the Mid line property I can assure that MN and AC are parallel.
So let's say this:
<CAN = <ANM
<ACM=<CMN
So with those two given, and the definition of median, I can deduce that
Triangle MGN is similar to Triangle AGC.
Reminding myself of the thales theorem that states, that if I draw a line parallel to two lines it will have all the points that divide the infinite lines in the same proportion.
I can say that BG will be twice the point that intersects AC.
And taking that point and N, qith the segment AB, and using that same theorem, I will conclude that point is the middle point of AC.
Concluding that the medians intersect in the "G" point.

No fomral proof stated here. But Since it's a existencial proof. We only have one point, so therefore These medians won't intersect anywhere else than "g".