Homothety, with centre O (a point) and ratio k (a number belonging to R\{0}) , is a geometric transformation where each transformed point Q' corresponds to an initial Q.

Q' belongs to the straight line passing through O and Q;
if k>0, then Q' is located on the same side as Q in relation to O;
if k<0, then Q' is located on the opposite side of Q in relation to O;
the ratio Q'O/QO is equal to |k|.

k is the scaling ratio and, as mentioned before, the value 0 for k is not acceptable.
If it was, each point of the plane would collapse to the centre of homothety. It would result in having an implosion. Fell free to test this in the interactive sample filling in the k-field with a 0, and clicking the button 'generate homothety'.

Essentially:

k>1 generates an enlargement;
k=1 is the identity;
0<k<1 generates a shrinking;
-1<k<0 generates a shrinking;
k=-1 generates a central symmetry;
k<-1 generates an enlargement;

Try all the values for k listed above in the interactive example to figure out what they exactly mean and produce in terms of transformation.
Furthermore you should take notice of the fact that in the sample the coordinates of the point O are not alterable, forcing the centre of homothety to overlap the origin of the whole system of coordinates.
This is simply a decision I made to simplify the programming. In a further update I'll implement the possibility to change the coordinates of the centre of homothety.

The example draws, in addition to the initial shape and the transformed one, the straight lines passing through the centre of homothety, the vertices of the triangular shape and their counterparts in the transformed shape. This aspect is based on the knowledge of the equation of a straight line passing through 2 points. Thank to this equation it is possible to properly draw the 3 straight lines of the sample computing for each its m (slope) and q (y-intercept).

If you like, try to have a look at this article which focus on straight lines: "Straight line".