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Drawing a Circle Tangent to an Angle

Date: 05/13/2000 at 18:49:35
From: Andy Klee
Subject: Drawing a circle within an angle and through a given point

I was asleep last year when my geometry teacher had this extra credit 
problem on the board and explained the answer. It has mystified me 
ever since. Please help!

Given an angle and any point inside the angle, not on the bisection 
line of the angle, you can draw a circle that goes through the point 
and is tangent to both sides of the angle. How do you do this with a 
compass and protractor?

I know how to bisect the angle and that the center of the circle must 
lie on the bisection line. But I don't know how to find the center 
point of the circle on this line.  

Thank you for your help!

Date: 05/15/2000 at 06:06:45
From: Doctor Floor
Subject: Re: Drawing a circle within an angle and through a given 

Hi, Andy,

Here is a figure with my construction:


We will suppose that D is further from AC than from AB.

Let E be the foot of the perpendicular altitude from D to AC. Draw the 
circle with center D through E. This circle is thick in my figure. I 
will call it the inversion circle. We will make use of "inversion" in 
this circle. There is no need for you to know this concept to follow 
the construction, but a bit of explanation is given in the Dr. Math 
archives at:

  Circumscribing Tangent Circles   

The inverse of AC (as a line, not a ray, but that is not important) is 
the circle with DE as a diameter (the midpoint F of DE is the center 
of this circle). The inverse of AB (again as a line) is found in the 
following way: intersect AB with the inversion circle at points H and 
G. The circumcircle of triangle HGD is the inverse we're looking for.

The inverses of the circles we're looking for must be tangent to the 
circles we just constructed. The inverses must become lines, because 
the circles pass through D, the center of the inversion circle.

So the inverses of the circles are the common tangents to the 
just-constructed circles. These tangents start from P in my figure.

Now we invert these two tangents back:

One tangent intersects the inversion circle in O and N, resulting in 
the circumcircle (C1) of triangle OND.

The other tangent intersects the inversion circle in L and M, 
resulting in the circumcircle (C2) of triangle LMD.

The two circles (C1) and (C2) are the circles we looked for.

Some basic construction skills you need to do the above construction 
can be found in the Dr. Math archives at:

  Geometry Constructions with Compass and Straightedge   

  Constructing the Orthocenter   

  Line Tangent to Two Circles   

If you have questions, or need more help, just write back!

Best regards,
- Doctor Floor, The Math Forum   

Date: 03/29/2004 at 12:56:15
From: Gwen
Subject: Geometry (construction)

In your answer to "Drawing a Circle Tangent to an Angle", you gave a 
very long example.  My math teacher had this same problem, and gave it 
to us for extra credit, and when someone gave him your answer, he said 
it was a different way than he was given, so there is another way. Could 
you give me the other way (he said it had something to do with dilations)?

Date: 03/29/2004 at 14:22:45
From: Doctor Peterson
Subject: Re: Geometry (construction)

Hi, Gwen.

That single word "dilation" gave me all the hint I needed!  It's a 
nice trick.

Suppose you just construct the angle bisector and choose an arbitrary 
point on it as the center of a trial circle tangent to the angle, 
which of course will not be the solution.  If D is the point given, 
construct the ray AD from the vertex of the angle; call one of the 
two intersections of this ray with the circle D'.  Now think about 
what will happen if you perform a dilation with center A that takes 
D' to D.  What happens to the circle?  How can you use that to 
construct the desired circle?

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
Associated Topics:
High School Conic Sections/Circles
High School Constructions
High School Geometry

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