James is constructing a circle circumscribed about a triangle. He has partially completed the construction, as shown below. What should be his next step in the construction?

Question

Triangle DEF is shown with two sets of intersecting arc markings on either side of side EF. A line segment is drawn through each set of intersecting arc markings and through side EF. Another segment is drawn through side DE and two more sets of intersecting arc markings.

Connect the arc markings to angle E to find another bisector.
Use the compass to find the perpendicular bisector for side DF.
Connect three arc markings to determine the vertices from the triangle.
Use the circumcenter to determine the center of the circle.

Answer 1

In the construction of a circumcircle for triangle DEF, James has already marked the intersections of arcs to find angle bisectors. The next step in the geometric construction would be to find the perpendicular bisector of one of the triangle's sides.

Specifically, he should:

Use the compass to find the perpendicular bisector for side DF.

Once he has completed the perpendicular bisectors for two sides (which he may have already started), he will be able to find the circumcenter of the triangle by determining where these perpendicular bisectors intersect. This circumcenter will serve as the center of the circumscribed circle.