Question

Points D, E, and F are not in a line. To construct a circle through points D, E, and F, begin by drawing line segments ("DE" ) ̅ and ("EF" ) ̅. Then construct the perpendicular bisectors of ("DE" ) ̅ and ("EF" ) ̅, and name the point of intersection of the perpendicular bisectors O. How do you know that point O is the center of the circle that passes through the three points?

Answer 1

Join all 3 points D, E, F, forming the triangle DEF.

Let the midpoint of EF be M and the midpoint of ED be N. (first picture)

Join point I to E, D and F.

Since IN is both an altitude and median to triangle EID, then triangle EID is an isosceles triangle, and IE=ID

similarly, we see that IE=IF.

conclusion: IE=ID=EF.

so O is the center of the circle with radius IE(=ID=EF)