A homothety is determined by fixing the center and the ratio $k\neq 0.1$. The image of a point $P$ of the plane by the homothety is obtained by multiplying by the ratio $k$ the vector that joins the point $P$ with the center:

$P'=O+k\vec{OP}$

 The only fixed point of the homothety is the center. 

Homotheties preserve angles. However, the distances are multiplied by the ratio $k$, while the areas are multiplied by the factor $k^2$

In the figure we see what is the image of a triangle by a dilation. You can check how the dilations behave in the plane, modifying the center, the ratio $k$ and the vertices of the triangle.

Anhomotthey in the plane, does it preserve orientation?