Homothety geometry

Question

I know that the product of $H(B, {1\over k})\circ H(A,k)$ is a translation ($H$ is homothety and $A,B$ are different points, $k$ is ratio), but what translation?

I think this is a translation, but what's the distance of this translation, any properties of this translation?

Answer 1

Well, pick any test point and find its image. The vector from the point to its image is the vector of translation.

Also, you can pick any test point, the outcome will be the same. But things may turn out to be way easier if your test point is $A$ or $B$. Probably $A$ is the most convenient (if you use the standard right-to-left composition).

UPDATE: what I said above is the easy way to determine the vector of translation, provided you already know that this composition is indeed a translation. If you need proof of that fact itself, I'll add it.

Answer 2

You actually want to use the test points A, and the point C on AB such that AC:AB =1:k. Then you can see that AC first stretches to AB and them to DB, where DB:AB=1:k. Hen it is a translation of K/ k-1 AB