Finding the Center of a Circle

Date: 5/29/96 at 15:3:49
From: Anonymous
Subject: Circle centers

How do you find the center of a circle, knowing nothing about it 
except its physical shape? Please help!

Date: 5/29/96 at 16:0:43
From: Doctor Pete
Subject: Re: Circle centers

If you have a straight edge (unmarked ruler) and a compass, it is easy 
to locate the center of a circle. Draw any line L through the circle, 
so it intersects the circle at A and B.  Then with the compass, find 
the perpendicular bisector of AB, which will intersect the circle at C 
and D. Clearly, CD is a diameter, so if you bisect CD at point O, O is 
the center of the circle.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 5/29/96 at 21:59:41
From: Anonymous
Subject: Finding the Center of a Circle

Thanks so much. Do you think this would work as well:
 
Take two parallel tangents to a circle. Draw the segment that runs 
through the points of tangency. This is the diameter. Is this true?

Date: 5/30/96 at 14:37:11
From: Doctor Darrin
Subject: Re: Finding the Center of a Circle

The line that you described would be a diameter.  To see this, we have 
to show that it passes through the center of the circle.  So, 
construct two lines, one passing through each point of tangency, and 
both passing through the center of the circle.  Both of these lines 
will be perpendicular to the tangent line at the point of tangency; 
since the tangent lines are parallel, the two lines that we construct 
are parallel.  However, they both pass through the same point (the 
center), so they must be the same line.  This line passes through both 
points of tangency and the center, so it is a diameter, since it is 
the unique line passing through the two points of tangency.  So we are 
done.

-Doctor Darrin,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    

Hi Dr. Pete, 

I thought you'd like to hear a real application of your solution.

I was trying to move a doorknob from an old door to a new one. When a 
doorknob is purchased new, it comes with a paper template that the builder 
can fold over the edge of the door. The template indicates exactly where 
to drill the big hole in the door by an X to mark the center of where the 
hole should be. But with an old doorknob, I had to create my own template 
from the old door. 

First I folded a blank paper over the edge of the old door and traced the 
outlines of the old hole. This gave me the circle to be drilled, but not 
the center. Then I used your method to mark the center of the circle. 
Then I folded the new template over the edge of the new door and drilled 
where I had marked. And it worked perfectly! 

Thanks for your solution!  And for the whole site. 

Warren S. Wolfeld