From: Tim Walters <twalters@retemail.outes>
Subject: Problem solved
Date: Sun, 16 Apr 2000 16:04:17 +0200
Newsgroups: sci.math
Summary: [missing]

I finally solved this problem myself. It's quite satisfying to work out
these thinglets, but it does waste a lot of time. In case anyone else finds
it interesting, my solution is as follows.

The Problem:

Given two circles (A) and (B), their radii and separation, and an intercept
crossing the axis between them at a known point and angle, we must
construct, with straight edge and compasses alone, the four circles centred
on the intercept and tangent to (A) and (B).

Solution:

Step 1: Construct the radical line (RL) of the two circles, and find their
homothetic centres (HC), one internal and one external. (If you don't know
how to construct RLs and HCs, do a web search, or check out Weisstein's
Concise Encyclopedia of Mathematics.) Let N be the point of interception of
the RL and the intercept.

Step 2: Construct perpendiculars from the HCs to the intercept. Take these
as polars, and find their poles (two per circle). (For polars and poles, see
Weisstein.)

Step 3: Draw rays from N to the four poles. These rays intersect the circles
at eight points (four pairs of points, each pair having its partner on the
other circle). Select one of these points, and construct a line passing
through it and the centre of its circle. The point where this line meets the
intercept is the centre of one of the four tangent circles. Constructing
this circle will identify the point's partner on the other circle. Repeat
this operation for the other three pairs of points.

(In the interests of clarity of expression and precision in language, I'd
welcome any suggestions as to how to make this explanation easier to
follow.)

Tim

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==============================================================================

From: Tim Walters <twalters@retemail.outes>
Subject: Re: Problem solved
Date: Sun, 16 Apr 2000 16:33:15 +0200
Newsgroups: sci.math

Tim Walters wrote:

> ... (four pairs of points, each pair having its partner on the
> other circle).

Sorry. That should read: ... (four pairs of points, each point having its
partner on the other circle).

Tim

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