How can be a line externally divide? We have studied the division of line internally and externally..
I just wondered how can a line be divided externally means with a point which is not on line? means what is the use to divide a line with an external point..
formula for internal division coordinates $$(x, y) = \left(\frac{m_1x_2+m_2x_1}{m_1+m_2}, \frac{m_1y_2+m_2y_1}{m_1+m_2} \right)$$
Can any one derive a formula for external division coordinates with figure?
Thanks!
 A: The terms are totally new to me, but here is what I learned from this explanation.
Both terms appear to always refer to the positioning of three collinear points. The "division" part apparently refers to the act of splitting up line segments between these three points.
The term "internal division" appears to mean the act of finding a point $R$ on the line segment $\overline{PQ}$ such that the lengths of $\overline{PR}$ or $\overline{RQ}$ have some prescribed property. For instance, $R$ might divide the segment in half, or in thirds, or in some other proportion. The "internal" part is derived from the fact $R$ is inside the segment.
"External division," on the other hand, apparently refers to locating a point $R$ collinear with $P$ and $Q$ but outside the segment $\overline{PQ}$, such that the lengths of segments again have some prescribed property. So, for example, one could ask "find a point $R$ on the line $\overleftrightarrow{PQ}$ such that the length of $\overline{PR}$ is twice that of $\overline{PQ}$." Such an $R$ could not lie within the segment $\overline{PQ}$, of course, so it is "external" to that segment.
These two actions are in the spirit of Euclidean geometry. Hilbert's axioms of "betweenness" include an axiom that says "For two distinct points $A$ and $B$ there is a third point $C$ such that $C$ is between $A$ and $B$." and another one that says "For distinct points $A$ and $B$, there is a third point $C$ such that $B$ is between $A$ and $C$." You can see that these are similar to what's going on above.
Internal and external division are evidently the metric geometry versions of these axioms, and they are stronger since they allow a lot of control over where the third point is placed. 
