Secant Line Calculator
Calculate secant-secant and secant-tangent relationships using circle theorems
Input Values
Formula: (a)(a+b) = (c)(c+d)
Where a, c are external segments and b, d are internal segments
Results
Visual Diagram
Secant-Secant Theorem:
(PA)(PB) = (PC)(PD)
or (a)(a+b) = (c)(c+d)
When two secants are drawn from an external point
Understanding Secant Lines
A secant line is a line that intersects a circle at two points. Secant theorems describe the relationships between secant segments and other circle elements when drawn from an external point.
Key Theorems
- Secant-Secant Theorem: When two secants are drawn from the same external point, the product of one secant and its external segment equals the product of the other secant and its external segment
- Secant-Tangent Theorem: When a tangent and a secant are drawn from the same external point, the square of the tangent equals the product of the secant and its external segment
- Power of a Point: All these products equal the same value, called the power of the point with respect to the circle
- These theorems are applications of similar triangles and proportional segments
Important Concepts
- External Segment: The portion of a secant from the external point to the near intersection with the circle
- Whole Secant: The entire length from the external point to the far intersection with the circle
- Internal Segment: The chord portion inside the circle (whole - external)
- Power of a Point: A constant value for all secants/tangents from the same point
- If the point is outside the circle, the power is positive
Applications
- Geometric constructions and proofs
- Finding unknown lengths in circle problems
- Optics and reflection calculations
- Computer graphics and collision detection
- Engineering design problems
- Mathematical competitions and problem solving
Frequently Asked Questions
What is a secant line?
A secant line is a straight line that intersects a circle at exactly two distinct points. It's different from a tangent line, which touches the circle at only one point.
What is the secant-secant theorem?
The secant-secant theorem states that when two secants are drawn from the same external point, the product of one secant and its external segment equals the product of the other secant and its external segment: (a)(a+b) = (c)(c+d).
What is the secant-tangent theorem?
The secant-tangent theorem states that when a tangent and a secant are drawn from the same external point, the square of the tangent length equals the product of the secant and its external segment: (tangent)² = (external)(whole secant).
What is the power of a point?
The power of a point is a value that remains constant for all secants and tangents drawn from that point to a circle. It equals the product calculations in both secant theorems and represents the point's "distance" from the circle.
How do you calculate the external segment?
The external segment is the distance from the external point to the nearest intersection of the secant with the circle. If you know the whole secant length and the chord length inside the circle, subtract the chord from the whole secant.
Can these theorems be used for points inside the circle?
The secant-secant and secant-tangent theorems specifically apply to external points. For points inside a circle, different theorems apply, such as the intersecting chords theorem where the products of the segments are equal.
What's the difference between a secant and a chord?
A chord is a line segment with both endpoints on the circle. A secant is a line (extending infinitely) that passes through the circle, intersecting it at two points. The portion of a secant inside the circle is a chord.
How are these theorems proven?
These theorems are proven using similar triangles. When you draw the appropriate radii and connect points, you can show that certain triangles are similar, leading to proportional sides that give the secant relationships.