Problem

Construct the circumcircle of right triangle \(DEF\), with right angle \(E\), using sides \(DE\) and \(EF\). State the \(\text{two}~\text{construction}~\text{lines}\) and their intersection \(O\), justify \(OD~=~OE~=~OF\), draw the circle from \(\text{one}~\text{vertex}~\text{radius}\), and identify \(O\)'s exact location on \(DF\).

Unworked right triangle DEF with right angle E and sides DE and EF named for the construction.

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Construct the perpendicular bisectors of DE and EF; their intersection O satisfies OD=OE=OF, so draw the circle centered at O through D. Because ∠E is right, O is the midpoint of hypotenuse DF.

Construct the angle bisectors of ∠D and ∠E; center the circle at their intersection and draw it through D.

Connect the midpoints of DE and EF; center the circle at that segment’s midpoint and draw it through D.

Construct the perpendicular bisector of DE; center the circle at the midpoint of DE and draw it through D.

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