Circle Theorems

Circle theorems reveal fascinating relationships between angles, arcs, and segments within circles. Understanding these theorems enhances problem-solving skills in geometry, connecting concepts from Elementary Algebraic Geometry and Honors Geometry to real-world applications and deeper mathematical insights.

1. Inscribed Angle Theorem

   • The measure of an inscribed angle is half the measure of the intercepted arc.
   • Inscribed angles that intercept the same arc are congruent.
   • This theorem helps in solving problems involving angles and arcs in circles.

2. Central Angle Theorem

   • A central angle is equal to the measure of the arc it intercepts.
   • Central angles that intercept the same arc are congruent.
   • This theorem is fundamental in understanding the relationship between angles and arcs in a circle.

3. Thales' Theorem

   • If A, B, and C are points on a circle where the line segment AC is the diameter, then the angle ABC is a right angle.
   • This theorem establishes a crucial relationship between diameters and angles in circles.
   • It is often used to prove other theorems related to right triangles inscribed in circles.

4. Tangent-Secant Theorem

   • The square of the length of the tangent segment from a point outside the circle is equal to the product of the lengths of the entire secant segment and its external segment.
   • This theorem is useful for solving problems involving tangents and secants from external points.
   • It provides a relationship between different segments associated with circles.

5. Chord-Chord Power Theorem

   • The product of the lengths of two segments of one chord is equal to the product of the lengths of two segments of another chord that intersect inside the circle.
   • This theorem is essential for solving problems involving intersecting chords.
   • It highlights the relationship between chords and their segments within a circle.

6. Tangent-Chord Theorem

   • The measure of the angle formed between a tangent and a chord through the point of contact is equal to the measure of the intercepted arc.
   • This theorem helps in finding angles related to tangents and chords in circles.
   • It is often used in conjunction with other circle theorems to solve complex problems.

7. Cyclic Quadrilateral Theorem

   • The opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) are supplementary (add up to 180 degrees).
   • This theorem is crucial for understanding the properties of quadrilaterals in circles.
   • It can be used to solve problems involving angles and sides of cyclic quadrilaterals.

8. Perpendicular Chord Theorem

   • If a diameter of a circle is perpendicular to a chord, then it bisects the chord.
   • This theorem establishes a relationship between diameters and chords in circles.
   • It is useful for solving problems involving perpendicular lines and bisected segments.

9. Intersecting Chords Theorem

   • When two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.
   • This theorem is important for solving problems involving intersecting chords.
   • It provides a way to relate the segments of chords that cross each other within a circle.

10. Tangent Line Theorem

    • A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
    • This theorem is fundamental in understanding the properties of tangents and their relationship with radii.
    • It is often used in problems involving tangents and angles in circles.