C(UV) denotes the circle with diameter UV.

T(P,QR) denotes the tangential distance |PS|,

where point P lies outside C(QR), point S

lies on C(QR), and PS is tangent to C(QR).

Let A, B, C, and D be distinct, collinear

points in that order.

Construct a point E on line AD such that

|EF| = T(E,AB) = T(E,CD) = |EG|

Denote O1E by b and O2E by c

b^2-r1^2=c^2-r2^2

b^2-c^2=r1^2-r2^2

(b-c)*O1O2=(r1-r2)*BC

(b-c)=((r1-r2)*BC)/O1O2

all the quantities on the right side are known:

r1-r2 difference of the radii

O1O2 distance between the circles' centers

BC =O1O2- (R1+R2)

**SO: **

**
****b-c ** IS CONSTRUCTABLE (several ways , inter alia similar triangles or intersecting chords in a circle etc).

**b-c and b+c define point E.**

**Very nice problem!!**

*Edited on ***October 3, 2010, 3:42 pm**