Construction: Center, Given Circle

There are two nice methods for constructing the center of a circle given the circle itself.

First method: Use right triangles. Draw any chord of the circle, and let its endpoints be $A$ and $B$. Then draw a line perpendicular to $AB$ passing through $B$, and let this line intersect the circle for the second time at $C$.

Now repeat this with another chord of the circle, say chord $DE$. We'll get two diameters, like this (shown in red):

Hooray! So, we've discovered that the intersection of the two diameters drawn is the center of the circle.

But there's another method too. Drumroll, please...

Second method: Use triangle circumcircles. Draw three distinct points on the circle, and call them $A$, $B$, and $C$. Now draw the perpendicular bisectors of $AB$ and $AC$. The intersection of the perpendicular bisectors is the circumcenter of $\triangle ABC$. Since the circle itself is the circumcircle of $\triangle ABC$, this means that the intersection of the perpendicular bisectors is in fact the point we are looking for.

Constructing the Center Given a Circle, Extended

Thanks to Dr. Po and Hongyi for the inspiration! This is simply an attempt to break down a variation of one of the methods in their article a little further, with motivation before the method is stated. Keep in mind that the optimal use of articles that I submit, in my opinion, is when a student tries something by themselves, and the teacher asks a simple, obvious question when he or she gets stuck. Let's get started!

What is a center?

A center is a point.

We can immediately identify a point on the circle.

What can we identify immediately?

It has to be arbitrary...

We just identify another arbitrary point on the circle.

What do we need to use our compass?

We need a center and a point on the circle.

What would be a drawn point to use as our center?

We can use $A$ or $B$, but we're going to choose $A$ from now on.

What would be a drawn point to use as a point on our circle?

The only remaining point. So our circle has center $A$ and contains $B$.

What is the point?

The point is the intersection of the two circles besides $B$.

How can we unclutter our diagram?

We get rid of the circle.

What do we strip from the circle before we send it away?

What is the property that we can take from the circle?

$AB=AC$.

What do three points remind you of?

Three points form a triangle.

What do we want?

We want the center of the given circle.

What is the circle in terms of $\triangle ABC$.

It is the circumcircle. What is the center?

What is the circumcenter in relationship to three points?

Oh wait, that would be redundant. What is the circumcenter in relationship to two points?

The circumcenter, which we'll call $O$, is such that $OA=OB$.

What is the locus of points $P$ equidistant from $A$ and $B$?

The locus is the perpendicular bisector. What type of geometric object is this?

It is a line. What determines a line?

Which two points do you notice are such that $PA=PB$?

We don't have $P$ yet, so we put the center of our compass on $A$. What would be the most reasonable drawn point that would work as a point on our circle?

We draw a circle with center $A$ containing $B$.

What else can we draw?

We draw a circle with center $B$ containing $A$.

What's special about two circles?

We can intersect them.

Do we know the perpendicular bisector now?

Yes!

How do we take advantage of the fact that $C$ lies on the circle with center $A$ containing $B$?

What can we draw?

Draw from previous experiences...

We just draw the same thing as the circle with center $B$ containing $A$, except on the other side. That is, the circle with center $C$ containing $A$.

Can you find anything that determines $O$?

Two non-parallel lines intersect at one point. As to the perpendicular bisectors of $AB$ and $AC$, they intersect at $O$.

We constructed the center of the given circle!