A homothety is a transformation of an affine space determined by a point $S$ called its center and a nonzero number $\lambda$ called its ratio, which fixes $S$, and sends each $M$ to another point $N$ such that the segment $SN$ is on the same line as $SM$, but scaled by a factor $ \lambda $.
A homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point $S$ called its center and a nonzero number $\lambda$ called its ratio, which sends
$$ M\mapsto S+\lambda {\overrightarrow {SM}} \text , $$
in other words it fixes $S$, and sends each $M$ to another point $N$ such that the segment $SN$ is on the same line as $SM$, but scaled by a factor $ \lambda $. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if $\lambda>0$) or reverse (if $\lambda<0$) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line $L$ is a line parallel to $L$.
In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.
In Euclidean geometry, a homothety of ratio $\lambda$ multiplies distances between points by $|\lambda|$ and all areas by $\lambda^2$. Here $|\lambda|$ is the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds $1$. The above-mentioned fixed point $S$ is called homothetic center or center of similarity or center of similitude.
If the homothetic center $S$ happens to coincide with the origin $O$ of the vector space ($S \equiv O$), then every homothety with ratio $\lambda$ is equivalent to a uniform scaling by the same factor, which sends
$$ {\overrightarrow {OM}}\mapsto \lambda {\overrightarrow {OM}} \text . $$
As a consequence, in the specific case in which $S \equiv O$, the homothety becomes a linear transformation, which preserves not only the collinearity of points (straight lines are mapped to straight lines), but also vector addition and scalar multiplication.
The image of a point $(x, y)$ after a homothety with center $(a, b)$ and ratio $\lambda$ is given by $\big(a + \lambda(x − a), b + \lambda(y − b)\big)$.
Source: Wikipedia
