I am struggling with this question from my introduction to Modern Geometry class.
$A$ is any point on the circle of similitude of two circles whose centers are $O$ and $O^{\prime} .$ Draw $A O$, and on it lay off $A O^{\prime \prime}=k \cdot A O^{\prime} .$ With $O^{\prime \prime}$ as center draw a circle whose radius is $k$ times that of circle $O^{\prime} .$ The circles $O$ and $O^{\prime \prime}$ are homothetic with $A$ as their center of similitude.
I have done a construction in Geogebra, and it appears to accomplish the condition. However, I am not sure if the proof is trivial or I am missing something. Advanced Euclidean Geometry from Roger Johnson states these (I still haven't proved these):
Two circles may be regarded as homothetic to each other in two ways:
This is my sketch proof approach:
Proof. By hyphotesis, we know $A O^{\prime \prime}=k \cdot A O^{\prime}$ lay off on $AO$, and we have a circle with $O''$ as center and $k\cdot O'E$ as radius. $\implies$ Here I suppose I should use the first part of the theorem stated above, however I am having troubles doing that... Suppose $BZ\parallel GH$...
I really appreciate any help! I am attaching the geogebra file: dropbox
