Use straightedge (no compass) to find the centers of two intersecting circles

Question

We can use the compass and straightedge to find the center of one circle.

We have proven we cannot find the center of a circle with straightedge alone (see: http://www.cut-the-knot.org/impossible/straightedge.shtml)

But if we are given two intersecting circles, can we find the centers of both circles?

Answer 1

For a chord $HG$ of the first circle, we can construct two parallel chords $EF$ and $IJ$ of the second circle. Since $EFIJ$ is an inscribed trapezoid, $LK$ is a perpendicular bisector of its base and passes through a circle centre. Two of those give you the centre.

Answer 2

COMMENT:

If we can draw common tangents without compass, then they intersect in P. We find mid point C of common chord AB and conect it to P. The centers of the circles are on PC and it's extension.

Now we draw perpendiculars from tangent point D and E , they intersect with PC in M and N which are centers of two circles.