the circumscribed circle
Given a triangle, can you find one circle that passes through all three of its corners at once? Yes — and there is exactly one. That circle is the circumscribed circle, and it hugs the triangle from outside, touching only the three vertices. The job is to locate its centre and radius with compass and straightedge.
The centre, called the circumcentre, is the one point equidistant from all three vertices. Recall that the perpendicular bisector of a segment is exactly the set of points equidistant from its two endpoints. So construct the perpendicular bisector of side AB and of side BC; their intersection O is equidistant from A, B, and C all at once. Open the compass to |OA| and draw the circle — it sweeps through B and C as well. The radius |OA| is the circumradius.
The circumcentre can sit inside the triangle (for an acute triangle), exactly on a side at its midpoint (for a right triangle, by Thales' theorem the hypotenuse is a diameter), or outside it (for an obtuse triangle) — a nice early surprise. Circumscribed circles also underlie inscribing regular polygons: place n equally spaced points around a circle and join them. Beware the common mix-up: the circumscribed circle goes through the vertices, while the inscribed circle sits snugly inside touching the sides.
For triangle ABC, build the perpendicular bisectors of AB and BC; they cross at the circumcentre O with |OA| = |OB| = |OC|. The circle of radius |OA| centred at O passes through all three vertices.
Two perpendicular bisectors meet at the circumcentre, the point equidistant from all three corners.
Do not confuse the circumscribed circle (through the vertices, centre = circumcentre from perpendicular bisectors) with the inscribed circle (touching the sides, centre = incentre from angle bisectors). For an obtuse triangle the circumcentre lies outside the triangle entirely.