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?

join all the 3 points "d, e, f," forming the triangle "def."

let the midpoint of "ef" be "m" and the midpoint of "ed" be "n". (it's in the first picture)

now, 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."

the conclusion : ie = id = ef.

now we know that o is the center of the circle with radius "ie" ( = id = ef)

(note : do refer the attachments ! )

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step-by-step explanation: