Construction: Circumcenter

Drawing a Geometry Diagram : Constructing the Circumcenter

Step 1 : Draw a nice, big, triangle and its midpoint of two sides.

Step 2 : Next, draw a perpendicular through the two midpoints. Their intersection is the circumcenter.

Step 3 : The circumcircle can be construct by drawing a circle with the center the circumcenter and passing through $A$ (or $B$, $C$).

Okay, so we want to create the circumcenter of a (non-degenerate) triangle $ABC$.

Well, let's think about this. The circle that goes through $A$, $B$, and $C$ has to go through $A$ and $B$, $B$ and $C$, and $A$ and $C$.

So, let's draw in the perpendicular bisector of $AB$.

Well, now what? Think for a moment.

Construct another perpendicular bisector! If one perpendicular bisector contains all of the centers of the circles that go through $A$ and $B$, and the other contains all of the centers of the circles of $B$ and $C$, then their intersection should contain the center of the circle that goes through $A$, $B$, and $C$, i.e. the circumcircle!

So, we draw the perpendicular bisector of $BC$

Now that we have the circumcenter ($H$), we can draw the circumcircle of $ABC$.

As you can see, we have successfully constructed the circumcircle (and circumcenter) of triangle $ABC$!