Geometry – Find Center of Externally Tangent Circle

circlesgeometrytangent line

I've been struggling to find a way to resolve the following problem:

Let $C_1$ a circle of center $V$ and of radius $r_1$.
Let $A$ and $B$ two points outside of $C_1$, and $L$ a line passing by them.
Let $W$ a point on $L$.
Let $C_2$ a circle of center $W$ passing through $A$ (radius $r_2$ is the distance between $A$ and $W$).

I am looking for the coordinate of the point $W$ so that $C_2$ intersects $C_1$ in a single point (that is that $C_1$ and $C_2$ are externally tangent).

The L line, C_1 and a variable C_2 depending on point W

What do you think? I tried to visualize a logic behind it using GeoGebra but couldn't figure it out. Right now what I'm doing in my code (this is for generative design) is increasing the distance between $A$ and $W$ by small steps until the distance $VW$ is equal to $r_1 + r_2$, but I'm looking for the elegant, geometrical solution!

Best Answer

Construct on $L$ a point $P$ such that $AP=r_1$. Center $W$ of the tangent circle is equidistant from $P$ and $V$, hence it can be found as the intersection between $L$ and the perpendicular bisector of $PV$.

Note that two choices are possible for $P$, one leading to an internally tangent circle, the other to an externally tangent circle.

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