I've got a situation in which I have 2 circles (C1 and C2) and i need to find the line equation for the line that is tangent to both of these circles.

So as far as i'm aware, given a single point (P1) and C2's point and radius it is possible to quite easily get 2 possible points of tangency for C2 and P1 to make 2 line equations. But as i don't have P1, only the knowledge that the point will be one of a possible 2 points on C1, i'm not sure how to calculate this.

I assume it will be something along the lines of getting the 2 tangent line equations of C1 that are equal to the same of C2.

Both circles can have any radius, they could be the same or they could be hugely different. They will also never overlap (they can still touch though). And I'm looking for the 2 possible internal tangents.

Oh, and also, visuals would be very helpful haha :)

Answer:

Let `O`

be the intersection point between the line through the centers and the tangent.

Let `d`

be the distance between the centers and `h1`

, `h2`

be the distances between `O`

and the centers. By similarity, these are proportional to the radii.

Hence,

```
h1 / h2 = r1 / r2 = m,
h1 + h2 = d,
```

giving

```
h1 = m d / (1 + m),
h2 = d / (1 + m).
```

Then the coordinates of `O`

are found by interpolating between the centers

```
xo = (h2.x1 + h1.x2) / d
yo = (h2.y1 + h1.y2) / d
```

and the angle of the tangent is that of the line through the centers plus or minus the angle between this line and the tangent,

```
a = arctan((y2 - y1)/(x2 - x1)) +/- arcsin(r1 / h1).
```

You can write the implicit equation of the tangent as

```
cos(a).y - sin(a).x = cos(a).yo - sin(a).xo.
```

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