Circle Tangent Calculator

Find tangent lines to a circle from an external point.

Circle Properties

External Point

Understanding Circle Tangents

A tangent line to a circle is a line that touches the circle at exactly one point, called the point of tangency. From any external point, exactly two tangent lines can be drawn to a circle.

Key Properties

  • A tangent line is perpendicular to the radius at the point of tangency
  • From an external point, two tangent lines of equal length can be drawn
  • The angle between a tangent and a radius is always 90°
  • Tangent segments from an external point to the circle are congruent

Formulas Used

Distance from external point P(x₁, y₁) to center C(h, k):

d = √[(x₁ - h)² + (y₁ - k)²]

Tangent length:

L = √(d² - r²)

Point of tangency calculation:

Using the right triangle formed by the center, external point, and point of tangency, we can find the two tangent points using trigonometric relationships.

Conditions

  • External point: The point must be outside the circle (distance > radius)
  • On the circle: Only one tangent line exists (the point itself is the tangency point)
  • Inside the circle: No real tangent lines exist

Applications

Tangent lines to circles are used in:

  • Optics: Light reflection and lens design
  • Engineering: Gear design and belt systems
  • Physics: Trajectory calculations and orbital mechanics
  • Computer Graphics: Smooth curve connections and rendering

Example: For a circle centered at (0, 0) with radius 5, and external point (10, 0), the two tangent points are approximately (2.5, 4.33) and (2.5, -4.33).

This calculator provides mathematical calculations for tangent lines to circles based on standard geometric formulas. Results are for educational and informational purposes. While we strive for accuracy, please verify important calculations independently.

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