Circle Constructions

Definition

Circle Constructions are the procedural methods used to create specific circles or tangent lines using only a straightedge and a compass. These constructions are based on the fundamental properties of equidistant points and the perpendicularity of radii to tangents.

Why It Matters

They provide a fundamental understanding of tangency and equidistance, which are the building blocks for more complex architectural and engineering layouts.

Core Concepts

• Tangents from External Points:
  1. Join the external point $P$ to the center $O$.
  2. Construct the midpoint $M$ of $OP$.
  3. Draw a circle with center $M$ and radius $MO$.
  4. The intersections of this new circle with the original circle are the points of tangency.
  • How to read: “The points P, O, and M, with an auxiliary circle centered at M with radius M O.”
  • Meaning: $P$ is external, $O$ is the given circle’s center, $M$ is the midpoint of $OP$; the auxiliary circle (diameter $OP$) meets the original circle at right angles, yielding tangent points where radius is perpendicular to tangent.
• Finding the Center: To find the center of an existing circle, construct the perpendicular bisectors of any two non-parallel chords. Their intersection is the center.
• Triangle Circles:
  • Circumscribed Circle: The center (circumcenter) is the intersection of the perpendicular bisectors of the sides. It passes through all three vertices.
  • Inscribed Circle: The center (incenter) is the intersection of the angle bisectors. It is tangent to all three sides.

Connected Concepts

• Fundamental Geometric Constructions
• Tangents
• Secants
• Points of Concurrence in a Triangle
• Circle Fundamentals