What you’ll learn
Construct/understand triangle circumcircle.
Key idea
Perpendicular bisectors of sides locate a center equidistant from vertices.
Equal distances to vertices require side perpendicular bisectors; equal side distances solve the incircle problem.
Worked example
Enter the exact value or ordered values in the format requested below.
Right triangle ABC has A=(-3, 2), B=(27, 2), C=(-3, 74), with a right angle at A. Construct its circumcircle. Call the intersection O. Enter bisectors=...; center=...; radius=...; through=...; name two sides for perpendicular bisectors and all three vertices.
- The perpendicular bisectors of AB and AC meet at O=(12, 38).
- OA^2=OB^2=OC^2=1521=39^2, so radius 39 reaches all vertices; the incenter I=(9, 14) instead has equal side distances 12.
Answer: bisectors=AB,AC; center=O; radius=OA; through=A,B,C
Quick Check
Choose which candidate representation satisfies the requirements of 'Construct/understand triangle circumcircle' and justify the choice.
Right triangle ABC has A=(0, 0), B=(84, 0), C=(0, 80), with a right angle at A. Candidate 1 uses side perpendicular bisectors at O=(42, 40) and radius OA. Candidate 2 uses angle bisectors at I=(24, 24) and radius IA. Candidate 3 uses midpoint of AB. Select a circumcircle plan with a vertex-distance justification.