Construct a circle centered at the circumcenter that passes through any vertex of this triangle. Why does this circle pass through the other 2 vertices of this triangle?
[b]Recall this theorem: [/b] [br][br]If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment. [br][br]From the screencast:[br]Since [i]D[/i] lies on the perpendicular bisector of [math]\overline{AB}[/math], it is equidistant from [i]A[/i] and [i]B[/i].[br]Since [i]D[/i] lies on the perpendicular bisector of [math]\overline{BC}[/math], it is equidistant from [i]B[/i] and [i]C[/i].[br]Since [i]D[/i] lies on the perpendicular bisector of [math]\overline{AC}[/math], it is equidistant from [i]A[/i] and [i]C[/i]. [br][br]This, [i]D[/i] is equidistant from [i]A, B, [/i]and [i]C[/i]. This means we can draw a circle centered at [i]D [/i]that passes through [i]A, B, [/i]and [i]C[/i]. This circle is called the triangle's circumcircle. Point [i]D[/i] is called the circumcenter of the triangle.