Construct a circle tangent to two given circles and to one of them at a given point Let's say circle $A$ and circle $C$ are given and we have to find circle $G$ passing through point $D$ on circle $C$. Center of this third circle, of course, will be on line $CD$ so that center $G$ is same distance from $D$ and from circle $A$. Then, we mark on line $CD$ point $E$ where $DE$ = radius of circle $A$. Then erecting perpendicular at midpoint of segment $AE$, we find desired center.
I wonder if there is better solution to the problem.

 A: This is not a "better" solution, but a more complete one:

Given a circle about $A$ through $B$ and a circle about $C$ through $D$, construct a circle about $D$ with radius $AB$, intersecting the line $CD$ at $E$ and $F$.
Let $G$ be the intersection of line $CD$ with the perpendicular bisector of $AE$
and let $H$ be the intersection of circle $A$ with line $AG$
nearer to $G$ if $D$ is between $E$ and $G$,
but let $H$ be the farther intersection if $E$ is between $D$ and $G.$
Then $G$ is equidistant from $A$ and $E,$
and therefore (since $DE = AH$) $G$ is equidistant from $D$ and $H$,
so $H$ is on the circle about $G$ through $D$ and the circle about $G$ is tangent to the circle about $A$ at $H$.
Similarly, let $J$ be the intersection of line $CD$ with the perpendicular bisector of $AF$ and let $K$ be the intersection of circle $A$ with line $AJ$
nearer to $J$ or farther from $J$ as $D$ is between $F$ and $J$ or $F$ is between $D$ and $J.$ Then the circle about $J$ through $D$ is tangent to circle $A$ at $K$.
In an exceptional case, one of the perpendicular bisectors may be parallel to $CD.$
In that case the tangent line at $D$ is also tangent to the circle about $A.$
Depending on the relative locations of $A, B, C, D,$ each circle that is produced may have external or internal points of tangency with both circles or may have an internal point of tangency with one circle and external tangency with the other.
Also note that the construction works even if the original two circles intersect or if one circle is entirely inside the other.
