292. The Circumcentre's Secret · Always equidistant from the vertices

Try to break it

Drag A, B, or C around. The circumcentre P always sits at the intersection of the three perpendicular bisectors and stays equidistant from all three vertices, so the circle through A, B, C tracks every move. Drag the three points onto a single line and the bisectors become parallel — P shoots off to infinity and the "circle" becomes the line.

How you build it

Construct the circumcentre using the perpendicular bisectors of the sides.

• Place point A as the first vertex of the triangle.
• Place point B as the second vertex of the triangle.
• Place point C as the third vertex of the triangle.
• Draw segment AB.
• Draw segment BC.
• Draw segment CA to complete triangle ABC.
• Construct the perpendicular bisector of side AB.
• Construct the perpendicular bisector of side BC.
• Mark the intersection point P of the two bisectors.
• Draw the circumcircle with center P passing through A.

The proof, step by step

Prove that the circumcentre is equidistant from the three vertices.

• P lies on the perpendicular bisector of AB.
• Any point on the perpendicular bisector of a segment is equidistant from its endpoints. So, PA = PB.
• Similarly, P lies on the perpendicular bisector of BC, so PB = PC.
• Therefore, PA = PB = PC. P is equidistant from all vertices.

Worked example

In triangle ABC, the circumcentre is P. If the distance from P to vertex A is 12 cm, what is the length of the segment PB?

The circumcentre P is equidistant from all vertices of the triangle. Since PA = 12 cm, PB must also be 12 cm.

• 6 cm
• 12 cm — correct
• 24 cm
• 18 cm