⭕ Circle Packing Touching

Circle packing problems involve multiple circles that touch each other. Master these configurations for efficient problem solving.

🎯 Recognition Cues

Key Triggers

  • Multiple circles - Look for problems with several circles
  • Tangency - Look for circles that touch each other
  • Equal radii - Look for circles with the same radius
  • Chains - Look for circles arranged in chains or patterns

Common Setups

  • Circles tangent to each other
  • Equal radius circles
  • Circles in chains or patterns
  • Circles tangent to lines

🧩 Solution Template

Step 1: Identify the Configuration

  • Determine how many circles are involved
  • Identify which circles touch which others
  • Mark given radii and relationships

Step 2: Apply Tangency Properties

  • Equal tangents: From external point, two tangents are equal
  • Power of a Point: Use for tangent relationships
  • Equal radii: Use for equal tangent lengths

Step 3: Set Up Equations

  • Use tangency properties to set up equations
  • Substitute known values
  • Solve for the unknown

Step 4: Verify

  • Check that all circles can actually touch
  • Ensure all radii are positive
  • Verify the final answer

🔍 Worked Example

Problem: Three circles with radii 2, 3, and 4 are externally tangent to each other. Find the radius of the circle that is externally tangent to all three.

Solution: Step 1: Identify configuration

  • Three circles with radii 2, 3, 4
  • All externally tangent to each other
  • Need to find radius of fourth circle

Step 2: Apply tangency properties

  • Use the formula for externally tangent circles
  • For circles with radii $r_1$, $r_2$, $r_3$ and $r_4$:
  • $\frac{1}{r_4} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{2\sqrt{r_1r_2 + r_2r_3 + r_3r_1}}{r_1r_2r_3}$

Step 3: Calculate

  • $r_1 = 2$, $r_2 = 3$, $r_3 = 4$
  • $\frac{1}{r_4} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{2\sqrt{2 \cdot 3 + 3 \cdot 4 + 4 \cdot 2}}{2 \cdot 3 \cdot 4}$
  • $\frac{1}{r_4} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} + \frac{2\sqrt{6 + 12 + 8}}{24}$
  • $\frac{1}{r_4} = \frac{13}{12} + \frac{2\sqrt{26}}{24}$
  • $\frac{1}{r_4} = \frac{13}{12} + \frac{\sqrt{26}}{12}$
  • $\frac{1}{r_4} = \frac{13 + \sqrt{26}}{12}$
  • $r_4 = \frac{12}{13 + \sqrt{26}}$

Step 4: Verify

  • Check that all circles can touch ✓
  • Radius is positive ✓
  • Answer makes sense ✓

Answer: $r_4 = \frac{12}{13 + \sqrt{26}}$

⚠️ Common Pitfalls

Pitfall: Wrong tangency setup

  • Fix: Make sure you understand which circles touch which others

Pitfall: Forgetting about equal tangents

  • Fix: From external point, two tangents are always equal

Pitfall: Wrong circle packing formula

  • Fix: Use the correct formula for the specific configuration

Pitfall: Forgetting about external vs internal tangency

  • Fix: External tangency means circles don’t overlap

🔗 Related Patterns

💡 Quick Reference

Circle Packing Properties

  • Equal tangents: From external point, two tangents are equal
  • Power of a Point: Use for tangent relationships
  • Equal radii: Use for equal tangent lengths

Common Configurations

  • Two circles: Simple tangency
  • Three circles: More complex tangency
  • Four circles: Advanced tangency
  • Chains: Circles in sequence

Solution Strategy

  • Identify: Look for circle packing configurations
  • Apply: Use tangency properties
  • Calculate: Solve for unknown radii or distances
  • Verify: Check that all circles can actually touch

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