Positional Relationships Between Two Circles by melia209209

This GeoGebra applet dynamically demonstrates the five positional relationships between two circles: separate (externally disjoint), externally tangent, intersecting, internally tangent, and contained (internally disjoint). Users can interactively adjust the radii R₁, R₂ of the two circles and the distance between their centers O₁O₂ to observe in real time how the relationship between the center distance and the sum or difference of the radii determines the relative position of the two circles. The construction includes labeled center points O₁ and O₂, visible radius segments, intersection points with annotations, and real-time distance calculations. This provides an intuitive visual representation of the criteria for determining positional relationships between circles in plane and analytic geometry. The applet is designed for middle school geometry instruction, allowing students to explore how changing parameters affects whether circles are far apart, touching at one point externally, crossing at two points, touching at one point internally, or one circle lying entirely within the other. Through direct manipulation of the geometric elements, learners can discover and internalize the key conditions: two circles are separate when the center distance is greater than the sum of the radii; externally tangent when it equals the sum; intersecting when it lies between the sum and the absolute difference of the radii; internally tangent when it equals the difference; and contained when the center distance is less than the absolute difference.