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 and . Then construct the perpendicular bisectors of and , 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?
When we join all three points (D, E, F) then it forms a Δ DEF.
Now, Let the midpoint of EF be M and the midpoint of ED be N and Join point I to E, D and F.
Because, IN is both an altitude and median to Δ EID, then Δ EID is an isosceles triangle, and IE = ID.
Similarly, we see that IE = IF.
So, we can conclude that,
Hence, O is the center of the circle with radius IE, ID or EF.
NOTE: See picture attached.