Cyclic Polygons

Definition

A cyclic polygon is a polygon that can be inscribed in a circle (meaning all of its vertices lie on the boundary of the circle).

Why It Matters

Cyclic polygons have highly symmetric properties, making them critical in coordinate geometry, trigonometry, and calculating areas of complex polygons.

Core Concepts

• Cyclic Quadrilateral Theorem: The opposite angles of any inscribed (cyclic) quadrilateral are supplementary (they sum to $180^\circ$).
  • How to read: “The measure of one hundred eighty degrees.”
  • Meaning: Opposite angles of a cyclic quadrilateral sum to a straight angle because their intercepted arcs sum to a full circle.
• Brahmagupta’s Formula: Computes the area of a cyclic quadrilateral using its semiperimeter $s$ and side lengths.

Connected Concepts

• Inscribed Angles
• Circle Fundamentals
• Quadrilateral Hierarchy
• Brahmagupta’s Formula