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Geometry Circle Theorems & Angle Rules Mastery Guide

Updated May 2026 • 11 min read • By CalcSolver Mathematics Team
Table of Contents

Circles possess remarkable geometric symmetries. Mastering circle angle theorems allows you to solve high school geometry proofs and competition problems effortlessly.

1. Inscribed Angle & Central Angle Theorem

• Inscribed Angle Rule: An inscribed angle is exactly HALF of the central angle subtending the same arc. • Formula: θ_inscribed = 1/2 · θ_central

5. Tangent-Secant Theorem & Intersecting Chords

• Intersecting Chords Rule: AE × EB = CE × ED • Tangent-Secant Theorem: PT² = PA × PB (where PT is tangent from point P)

6. Secant-Secant Theorem & Power of a Point

• Power of a Point: For circle with radius R and point P at distance d from center, Power = d² - R². • Secant-Secant Rule: PA × PB = PC × PD (where secant lines intersect at external point P).

7. Area of Segment & Circular Sector Formulas

Sector Area A = (1/2) r² θ (rad). Segment Area = Sector Area - Triangle Area = (1/2) r² (θ - sin θ).

8. The Alternate Segment Theorem

The Alternate Segment Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment subtended by that chord.

9. Nine-Point Circle & Euler Line in Triangles

In any triangle, the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices all lie on a single circle known as the Nine-Point Circle!

10. Tangent Properties & Perpendicular Radii

• Radius-Tangent Orthogonality: A radius drawn to the point of tangency always forms a 90° right angle with the tangent line. • Tangents from External Point: Tangent segments from a common external point P to a circle are equal in length (PA = PB).

11. Cyclic Quadrilateral Ptolemy's Theorem

For any four points on a circle forming a cyclic quadrilateral ABCD, Ptolemy's Theorem states that the product of the diagonals equals the sum of the products of opposite sides: AC × BD = (AB × CD) + (BC × AD).

12. Solving Complex Geometry Competition Problems

Worked Example: Intersecting Chords

Chords AB and CD intersect at E inside a circle. AE = 4, EB = 6, CE = 3. Find ED.

By Intersecting Chords Theorem: AE × EB = CE × ED ⟹ 4 × 6 = 3 × ED ⟹ 24 = 3 ED ⟹ ED = 8.

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Frequently Asked Questions (FAQs)

What is Thales' theorem?
Thales' theorem states that any angle inscribed in a semicircle is always a 90-degree right angle.