Definition of Circumcircle and Circumcenter

The circumcircle of a triangle is a circle that passes through all of the vertices of the triangle. The circumcircle of a polygon is a circle that passes through all of the vertices of the polygon.

The circumcenter of a triangle is the center of the circumcircle of the triangle.

Constructing the circumcircle of a triangle (interactively!)

To construct the circumcircle of a triangle, follow these steps:

• Step 1. Construct the perpendicular bisectors of the sides of the triangle. (It is sufficient to construct any two of them, since all three intersect at a point.)
• Step 2. Mark the circumcenter, the point of intersection of the perpendicular bisectors.
• Step 3. Draw a circle with center at the circumcenter and passing through any of the vertices of the triangle.

You can view the steps in the following applet. Drag any of the vertices of the triangle to change its shape.

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A Closer Look at the Construction of the Circumcircle — Why Does It Work?

In the applet below, M is the midpoint of side AB and point O is the intersection of the three perpendicular bisectors of triangle ABC. The circle with center O and radii OA and OB has been constructed.

• Triangles OAM and OBM are congruent. Why?
• What does this mean about segments OA and OB?
• What about segment OC?
• What does this prove about the circle?

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Answer

Triangles OAM and OBM are congruent by Side-Angle-Side. Sides AM and BM are congruent, side OM is shared, and the included angle in each triangles is a right angle. This means that segments OA and OB are equal and the circle centered at O passing through vertex A must also pass through vertex B. A similar argument applies to segment OC, so vertex C is also on the circle, and this must be the circumcircle of the triangle.

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