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 Equal Tangents (Posted on 2010-10-03)
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|

 See The Solution Submitted by Bractals Rating: 3.0000 (1 votes)

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 Construction | Comment 4 of 11 |
This description may leave a little to be desired but I believe that it fulfils the requirements:

Let the centre of circle AB be A'.
Let the centre of circle CD be C'.

Construct square XYCB.

Join A' to X intersecting circle AC at F.
Join C' to Y intersecting circle CD at G.

Join F to G.

Construct a perpendicular bisector of FG to intersect line BC at E.

|FE|  =  |GE|.
 Posted by brianjn on 2010-10-04 02:20:35

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