MATHEMATICS LAB · GEOMETRY

Study of Circles

Edit equations and drag centers and point P on the Cartesian plane. Explore intersections, tangencies, and relative positions, with step-by-step calculations.

Radii must be positive, and coordinates are measured in arbitrary units (u). The lab distinguishes cases with no real solutions, tangencies, and coincident circles.

1. Define Circle C₁

Coordinates or equation coefficients

Equation form

Examples:

Display options

Controls and information

Grid Centers, radii, and constructions

Drag O₁ to move the circle, R to change the radius, and P to update the tangents. When C₂ is enabled, you can also drag O₂. Use +, −, or the mouse wheel to zoom and Fit to frame the entire construction.

C₁ and O₁ C₂ and O₂ Line ℓ Tangents from P Intersection points

3. Additional options: line, second circle, and tangents

Enable constructions to add to the graph

Intersection with a lineEquation of line ℓ

Second circle C₂

Center h₂Center k₂Radius r₂

Tangents to circle C₁ from point P

P x-coordinateP y-coordinate

Center O₁, radius R, center O₂, and point P are draggable on the graph. All input fields update automatically.

2. Interactive circle graph

Axes at a 1:1 scale

C₁ · main circle

4. Results

Updated values

5. How to construct intersections

Geometric interpretation

6. Explained calculations

Formulas and numerical substitutions

What you can explore in this version

Equations and representations

Standard and general equations, center, radius, real or impossible circles, and interactive moving and resizing.

Intersections and tangents

Secant, tangent, or external lines; tangents from a point inside, on, or outside the circle.

Two circles

Intersecting circles, internally or externally tangent circles, disjoint, concentric, and coincident circles.

The lab uses double-precision real numbers. Classifications near tangency use a numerical tolerance; approximate results do not replace formal mathematical proofs.