What if the points are in the same segment? Draw a different circle, with points \( P, Q \) and \( R \) in the same segment

1. Begin by choosing three distinct points \( P \), \( Q \), and \( R \) so that when later placed on a circle they will all lie in one circle‐segment (that is, they lie on the same side of an appropriate chord). 2. Draw the chord joining \( P \) and \( Q \). This chord will divide any circle passing through \( P \) and \( Q \) into two segments. We want to work with the segment (usually the “smaller” one) that will also contain \( R \). 3. To construct the desired circle, find the perpendicular bisector of the line segment \( PQ \). This line consists of all points that are equidistant from \( P \) and \( Q \). 4. Next, construct the perpendicular bisector of the segment joining one of the endpoints, say \( P \), and \( R \); again, this bisector represents all points that are equidistant from \( P \) and \( R \). 5. The intersection of these two perpendicular bisectors gives the center \( O \) of the unique circle passing through \( P \), \( Q \), and \( R \). 6. With center \( O \) determined, use a compass set to the distance \( OP \) (which equals \( OQ \) and \( OR \)) to draw the circle. 7. Verify that all three points lie on the circumference and, in particular, that \( R \) is located in the same segment determined by the chord \( PQ \) (i.e. \( R \) lies on the arc connecting \( P \) and \( Q \) that does not contain the other “major” arc). 8. If desired, label the arc between \( P \) and \( Q \) which contains \( R \) as the segment in question. This construction produces a circle in which the three points \( P \), \( Q \), and \( R \) lie together in the same segment.