# Ask Uncle Colin: Find me a circle!

Dear Uncle Colin,

I have two points and I want to construct a circle of a given radius that passes through them. Is it possible?

-- Every Underspecified Circle Lives Its Dream

Hi, EUCLID, and thanks for your message!

There are three possible answers to this, depending on the size of the radius (let’s call it $r$) and the distance between the points.

The easy one first: if the distance between the two points is greater than $2r$, then forget it - there’s no such circle.

Second, the next-simplest one: if the distance between the points is exactly $2r$, there’s one possible circle; its centre is the midpoint of your original points.

Thirdly, the one I suspect you meant: if the distance is smaller than $2r$, there are two possible circles. Draw a circle of radius $r$ around each of your points; these will intersect at two points. Each of these is the centre of a circle of radius $r$ passing through your points.

Hope that helps!

-- Uncle Colin