> 
> One of our users' questions goes like this:
> 
> Rob - 11:14pm Sep 24, 1996 (#3 of 4)
> 
> I've been asked to "construct" a circle that is tangent to both rays of an
> arbitrary angle and passes through an arbitrary point located "inside" the
> angle but *not* on the angle's bisector. I know that the center must be on
> the angle's bisector and that once the center is found, the points of
> tangency will be perpendicular to the rays of the angle and passing through
> the center. How can you find the location of the center using only a compass
> and straightedge. My teacher stressed that guessing is not a construction
> technic. 
> 
> (Ouch! My head hurts!)

It is impossible to construct a circle tangent to both rays of an 
arbitrary angle without the circle including exactly two points on 
the angle's bisector. The center of the circle will also be on the 
bisector. 

Mark off two points on the rays. Each should be an identical distance
(probably the width of the straightedge) from the origin. Using the
end of the straightedge as a 90 degree protractor, draw lines
perpendicular to each of those points. Move the straightedge and use
it to extend the lines until they intersect. The intersection will
be both the center of the circle and one point on the bisector of the
angle. The origin of the angle will form the other point on the
bisector. The circle may then be constructed with a compass by using
the perpendiculars' intersection as the center and the point where one
perpendicular intersects with one ray as the radius.

Now go construct the geometric proof that this is true.
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