> Let A, B be the centers of two circles with radius > r1, r2 respectively, where r1-r2 is very small. > Construct the exterior tangent to the given circles. > > In this puzzle is sought a practical construction on an A4 sheet of paper that avoids entirely the use of r1-r2.
Good puzzle Avni, Thanks. It may not be the construction you meant, but I found one that works.
I'll borrow Peter's nomenclature: 1) Start with line AB and circles C1(A,r1) and C2(B,r2). 2) Bisect AB to make point D. 3) Construct length (r1 + r2)/2 and call it r3. 4) Construct circle C3(D,r3). 5) Draw a line between the two points where C1 and C3 cross. Call the point where this line crosses AB point E. 6) Draw a line between the two points where C2 and C3 cross. Call the point where this line crosses AB point F. 7) Bisect line EF to make point G. 8) construct a line from point G perpendicular to line AB (also EF) that intersects circle C3. Label this intersection point H. 9) Point H is the midpoint of the desired tangent line. Construct a line from point H tangent to circle C1 in the standard way. 10) Extend this line to contact circle C2. It will be tangent.
This construction will work as long as length AB < 2(r1 + r2).
This can be simplified slightly if length AB < r1 + r2. Steps 5,6, & 7 can be eliminated. A line through the two points where C1 and C2 cross can substitute for line GH.
This construction is slightly more difficult when AB > 2(r1 + r2). 4A) Consider length DB as r4. 4B) Construct C4(D,r4). 5) Same as 5 above except where C1 and C4 cross. 6) Same as 6 above except where C2 and C4 cross. Remainder the same as above.
I have not yet proved this construction mathematically. I have, however, tested it sufficiently with a high level CAD program to be sure it is correct.