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Topic: Circle-circle tangent puzzle
Replies: 5   Last Post: Jun 17, 2012 5:24 AM

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Posts: 204
Registered: 12/6/04
Re: Circle-circle tangent puzzle
Posted: Jun 15, 2012 9:06 PM
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> 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.

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