Tangent Circumference Construction

Question

"Construct a circumference that is tangent to a given circumference and tangent to a line $r$ through a point $A$ of this line."

I've done the line perpendicular to $r$ through $A$, cause we know that the center of the circumference must lie in there (since the radius of the circumference must be perpendicular to $r$), but I don't know how to proceed...
Need some help! Thanks

Answer 1

Put the diagram on a Cartesian grid so that the circle center $C$ is at point $(0,d)$, the circle's radius is $r$, and point $A$ is at $(0,a)$. We are looking for point $D$ at $(a,y)$ such that the circle centered at $D$ with radius $y$ is tangent to the original circle.

Then the distance between points $C$ and $D$ equals $y+r$, so we get

$$\sqrt{(a-0)^2+(y-d)^2}=y+r$$

Solving this, we get

$$y=\frac{a^2+d^2-r^2}{2(d+r)}$$

This can be constructed, thus also point $D$.

Here is the construction:

Given are point $A$ on line $a$ and point $C$ center of circle $f$ (all in red).

Construct line $c$ through point $A$ perpendicular to line $a$. Construct a circle $j$ with diameter $\overline{AC}$. Mark point $F$ at the intersection of circles $f$ and $j$. Mark point $H$ at the intersection of circle $j$ and the line through point $C$ perpendicular to line $a$.

Draw circle $p$ centered at point $A$ going through point $F$. Mark the intersection $G$ of circle $p$ with the line $c$. Draw ray $\overrightarrow{AF}$. Find point $J$ on ray $\overrightarrow{AF}$ such that $AJ=2OH$. Draw segment $\overline{JG}$. Construct line $n$ through point $F$ parallel to line $JG$.

Mark the intersection $D$ of line $j$ with line $c$. Draw the circle $h$ centered at point $D$ going through point $A$. (All this in blue). This is your desired circumference!