To construct the perpendicular bisector of

*AB*:

- Draw two circles with the same radius and with centers at the endpoints of segment
*AB*. The radius must be long enough for the two circles to intersect. - Mark the points of intersection
*P* and *Q* of the two circles. - Draw line
*PQ*. This is the perpendicular bisector of segment *AB*. - Mark the intersection
*M *of line *PQ* with segment *AB*. This is the midpoint of segment *AB*.

Why does this construction work?

Click on the **Show Segments** button. This will create four line segments, *AP*, *AQ*, *BP*, and *BQ*.

- Why are segments
*AP*, *BP*, *AQ*, and *BQ* congruent? - What can you say about triangles
*APQ* and *BPQ*? Why? - Triangles
*PMA* and *PMB* are congruent. Why? - What does this imply about segments
*AM* and *MB*? - What is the angle between line
*PQ* and segment *AB*? Why? - What can you conclude?

See how to use perpendicular bisectors to construct the circumcircle of a triangle.