Find Snap Tangent Points

A common feature of CAD programs is support for tangent object snaps.  A line is drawn from some point, A, toward a circle with center C and radius r.  We want to find a point (or points), B, on the circle, such that AB is tangent to the circle.

Here's a sketch of the problem:

We are given the points A and C and the radius r.  The goal is to find B₁ and B₂.  We argue mainly for B₁. The argument for B₂ is the same.  We have assumed A is outside of the circle or else there are no solutions. (If you write code for this, you will need to check for this condition.)

Since AB₁ is tangent to the circle and B₁C is the radius of the circle, angle AB₁C is a right angle.  The distances f and from A to C, say, d, may be calculated using the pythagorean theorem.  Now, we draw the altitude of triangle AB₁C to E.  By similar triangles (ΔAB₁C ~ ΔB₁EC~ ΔAEB1) we determine that

and

and so, e = r²/d and a = f r/d.  We let δ be the normalized direction vector from C to A, namely,

Then point E is given by E = C + e δ.  Let δ_p= (δ_y, –δ_x), or the clockwise 90° rotation of δ.  Then points B₁ and B₂ are given by:

       B₁ = E + a δ_p
and
       B₂ = E – a δ_p,

which are the desired points.