External division of a line segment

[Nero26]

Suppose AB is a straight line segment. To divide the segment externally at point C in the ratio m:n, is it necessary for C to remain always outside of AB? I always thought C should be outside of AB, but the post in the link below confuses me
http://www.transtutors.com/math-homework-help/vectors/section-formula.aspx
Basically what is meant by external division of line segment by a point?
Can anyone please clarify me? Thanks.

 

[SteamKing]

I'm not sure what you are asking. In the link, the point C is clearly on the line segment between A and B.
The point O is external to the line segment.

 

[haruspex]

I would guess it means that the distances AC, BC should be in the ratio m:n. If that represents AC:BC (as opposed to the other way around) then there are two solutions, one internal and one external.

 

[Nero26]

Thanks for your response.

I'm not sure what you are asking. In the link, the point C is clearly on the line segment between A and B.
The point O is external to the line segment.

I'm trying to know if C is on line segment between A and B (fig 2) then can it divide AB externally or not? In the link it says C divides AB externally in the ratio AC:AB =m:n, is it correct or not?

Sorry for not stating clearly.

I would guess it means that the distances AC, BC should be in the ratio m:n. If that represents AC:BC (as opposed to the other way around) then there are two solutions, one internal and one external.

In (fig 1) if C divides AB externally then which pair of segments should we consider AC:CB or AC :AB?
Or does this consideration of segments depend on given ratios? (for m+n:n, AC:CB for m+n:m, AC:AB)

 

[haruspex]

In (fig 1) if C divides AB externally then which pair of segments should we consider AC:CB or AC :AB?

In the construct "C divides AB", A and B have symmetric roles, whereas C is different. Therefore it's AC:CB.

 

[Nero26]

In the construct "C divides AB", A and B have symmetric roles, whereas C is different. Therefore it's AC:CB.

What have you meant by "A and B have symmetric roles" here?
Please explain a little more. And what about the second fig, that if C is in between A and B on AB, then can it divide AB externally?
Sorry for bothering you.

 

[haruspex]

What have you meant by "A and B have symmetric roles" here?

"C divides AB" is the same statement as "C divides BA". The roles that B and A play in the statement are symmetric, so the two are interchangeable. In the ratio expression AC:AB, B and A are not interchangeable; AC:AB is different from BC:BA. Therefore the correct interpretation is the ratio AC:BC.

what about the second fig, that if C is in between A and B on AB, then can it divide AB externally?

If you compare the vertex labels A, B and C with the vector labels a, b, c you will see that there's an error in the diagram.

 

[Nero26]

Thanks a lot haruspex :)