How To Find the Circumcenter, the Center of the Circumscribed Circle of a Triangle

• The perpendicular bisector of a segment

• The perpendicular bisector of a segment is the line perpendicular to the segment, that passes through the middle of it.
• Important: Any point belonging to the perpendicular bisector is equidistant from the ends of the segment.

• The perpendicular bisectors of triangle's sides

• In any triangle the perpendicular bisectors of the sides are concurrent in a point which is the center of the circle passing through the vertices of the triangle, called the circumscribed circle of the triangle (the circumcircle).
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  The perpendicular bisectors and the center of the circumcircle

• In the triangle △ABC, OM1 is the perpendicular bisector of segment BC =>
  BM1 = CM1 = ^BC / ₂ = ^a / ₂. The length of segment BC = a.
• OM2 is the perpendicular bisector of segment AC =>
  AM2 = CM2 = ^AC / ₂ = ^b / ₂. The length of segment AC = b.
• OM3 is the perpendicular bisector of segment AB =>
  AM3 = BM3 = ^AB / ₂ = ^c / ₂. The length of segment AB = c.

• The perpendicular bisectors OM1, OM2 and OM3 intersect at point O, which is the center of the circumscribed circle of the triangle.

• Why are the perpendicular bisectors concurrent in the center of the circumscribed circle?

• O is a point on the perpendicular bisector OM1 => BM1 = CM1 = ^a/₂; OM1 ⊥ BC (⊥ is the symbol for segments or perpendicular lines); and OM1 is a common segment => the right triangles △OBM1 ≅ △OCM1 => OB ≅ OC.
• O is a point on the perpendicular bisector OM2 => AM2 = CM2 = ^b/₂; OM2 ⊥ AC; and OM2 is a common segment => the right triangles △OAM2 ≅ △OCM2 => OA ≅ OC.
• From the relations above, it follows that OA = OB = OC = R, where R is the radius of the circumscribed circle of the triangle.

» How To Calculate the Radius of the Circumscribed Circle of a Triangle Given the Lengths of the Triangle's Sides