Does the formula we just saw look vaguely familiar? Okay! It may not look familiar, but it does have many of the components of another formula we have seen.
Since we have been dealing with similar triangles, the concept of dilations may come to mind.
Formula for a dilation, center not at the origin:
O = center of dilation at (a,b); k = scale factor
Regarding directed line segment , we will be dilating the endpoint B using the endpoint A as the center of the dilation. Since our partition point lies ON the segment, we will be dealing with a dilation which is a reduction (0 < k < 1).
The image of the dilation will be the partition point, P.
The variables a and b do not represent the same quantities in the two formulas we are examining, so if we tweek this dilation formula to correspond to our given information, we may see the resemblence between the formulas more clearly.
Remember, the point being dilated is B, and its image will be the partition point P.
The center of our dilation is point A at (x_{1}, y_{1}), so we will replace a with x_{1} and b with y_{1}.
The point being dilated is B, so we will replace x with x_{2} and y with y_{2}.
If we rearrange a couple of terms (commutative property), we will get:
Let's compare this improved dilation formula to our previous formula for finding P:
Notice:
The scale factor of the dilation, k, is equal to the ratio of AP to the total length AB.
If you choose to use this dilation method when partitioning segments, remember that the "partition ratio" (2/3 in this case) is NOT the scale factor. The "AP distance ratio" representing a portion of the whole segment length AB (2/5 in this case) is the scale factor. 
