Secant-Tangent Power Calculator
Power of a Point Theorems (External Point)
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Secant 1 (from external point)
Secant 2 (from same external point)
Two Secants Theorem
(whole₁)(ext₁) = (whole₂)(ext₂)
Diagram
Power of a Point (External)
When lines from an external point intersect a circle, the products of their segments have a special relationship. This is the "power of a point" theorem for external points.
The Three Cases
Two Secants
(whole₁)(ext₁) = (whole₂)(ext₂)
Both lines pass through the circle
Secant + Tangent
(whole)(ext) = tangent²
One secant, one tangent
Two Tangents
tangent₁ = tangent₂
Both lines are tangent
Understanding the Formula
For a secant from external point P:
- Whole segment: From P to the far intersection
- External part: From P to the near intersection
- Internal part: The chord inside the circle (whole - external)
Power of Point Formula
For an external point at distance d from center of a circle with radius r:
Power = d² - r²
Frequently Asked Questions
What is the "whole" vs "external" segment?
The "whole" is the entire length from the external point through both circle intersections. The "external" part is just from the external point to the first (closer) intersection.
Why does this work for tangents?
A tangent can be thought of as a secant where both intersection points coincide. So (whole)(external) becomes tangent × tangent = tangent².
Why are two tangents from the same point equal?
The two tangent segments, along with radii to the tangent points, form two congruent right triangles. Their hypotenuses (the tangent segments) are therefore equal.
How does this differ from intersecting chords?
For chords intersecting inside: (part₁)(part₂) = (part₃)(part₄). For external point: we use (whole)(external), not individual parts.
Can I find the radius from power of a point?
Yes! If you know the distance d from external point to center and the power: r = √(d² - power).