Consider the figure on the right showing two circles with radii r₀ and r₁. The points p₀, p₁, p₂, and p₃ have coordinates (x₀, y₀) and so forth.

Let d = the distance between the circles' centers so . Solving for a gives .Now there are three cases:

• If d > r₀ + r₁: The circles are too far apart to intersect.
• If d < |r₀ - r₁|: One circle is inside the other so there is no intersection.
• If d = 0 and r₀ = r₁: The circles are the same.
• If d = r₀ + r₁: The circles touch at a single point.
• Otherwise: The circles touch at two points.

The Pythagorean theorem gives:

So:

Substituting and multiplying this out gives:

The -b² terms on each side cancel out. You can then solve for b to get:

Similarly:

All of these values are known so you can solve for a and b. All that remains is using those distances to find the points p₃.

If a line points in direction <dx, dy>, then two perpendicular lines point in the directions <dy, -dx> and <-dy, dx>. Scaling the result gives the following coordinates for the points p₃:

Be careful to notice the ± and ∓ symbols.

Click and drag to create two circles on the example program. The following code shows the FindCircleCircleIntersections method that the program uses to find the intersections.

' Find the points where the two circles intersect.
Private Function FindCircleCircleIntersections( _
    ByVal cx0 As Single, ByVal cy0 As Single, ByVal radius0 _
        As Single, _
    ByVal cx1 As Single, ByVal cy1 As Single, ByVal radius1 _
        As Single, _
    ByRef intersectionx1 As Single, ByRef intersectiony1 As _
        Single, _
    ByRef intersectionx2 As Single, ByRef intersectiony2 As _
        Single) As Integer
Dim dx, dy As Single
Dim dist, a, h, cx2, cy2 As Double

    ' Find the distance between the centers.
    dx = cx0 - cx1
    dy = cy0 - cy1
    dist = Sqr(dx * dx + dy * dy)

    ' See how many solutions there are.
    If (dist > radius0 + radius1) Then
        ' No solutions, the circles are too far apart.
        intersectionx1 = NAN
        intersectiony1 = NAN
        intersectionx2 = NAN
        intersectiony2 = NAN
        FindCircleCircleIntersections = 0
        Exit Function
    ElseIf (dist < Math.Abs(radius0 - radius1)) Then
        ' No solutions, one circle contains the other.
        intersectionx1 = NAN
        intersectiony1 = NAN
        intersectionx2 = NAN
        intersectiony2 = NAN
        FindCircleCircleIntersections = 0
    ElseIf ((dist = 0) And (radius0 = radius1)) Then
        ' No solutions, the circles coincide.
        intersectionx1 = NAN
        intersectiony1 = NAN
        intersectionx2 = NAN
        intersectiony2 = NAN
        FindCircleCircleIntersections = 0
    Else
        ' Find a and h.
        a = (radius0 * radius0 - _
            radius1 * radius1 + dist * dist) / (2 * dist)
        h = Sqr(radius0 * radius0 - a * a)

        ' Find P2.
        cx2 = cx0 + a * (cx1 - cx0) / dist
        cy2 = cy0 + a * (cy1 - cy0) / dist

        ' Get the points P3.
        intersectionx1 = CSng(cx2 + h * (cy1 - cy0) / dist)
        intersectiony1 = CSng(cy2 - h * (cx1 - cx0) / dist)
        intersectionx2 = CSng(cx2 - h * (cy1 - cy0) / dist)
        intersectiony2 = CSng(cy2 + h * (cx1 - cx0) / dist)

        ' See if we have 1 or 2 solutions.
        If (dist = radius0 + radius1) Then
            FindCircleCircleIntersections = 1
        Else
            FindCircleCircleIntersections = 2
        End If
        Exit Function
    End If
End Function