🔄 Inversion & Spiral Similarity

These advanced transformation techniques appear occasionally in AMC 12 problems. They can provide elegant solutions to complex geometric configurations.

🎯 Key Concepts

Inversion (Very Light)

Definition: Transformation that maps points to their inverses with respect to a circle Properties:

  • Maps circles to circles or lines
  • Preserves angles
  • Useful for tangent circle problems

Basic Formula: If $P$ inverts to $P’$ with respect to circle of radius $r$ centered at $O$: $OP \cdot OP’ = r^2$

Spiral Similarity

Definition: Combination of rotation and homothety (scaling) about the same center Properties:

  • Preserves angles
  • Scales distances by constant factor
  • Useful for similar figures

Key Insight: Spiral similarity can map one figure to another similar figure

🔍 Micro-Examples

Inversion Example

Point $P$ at distance 5 from center of circle of radius 3:

  • Inversion distance: $OP’ = \frac{3^2}{5} = \frac{9}{5} = 1.8$

Spiral Similarity Example

Rotate triangle by $60°$ and scale by factor 2:

  • This is a spiral similarity
  • All distances double, all angles preserved

⚠️ Common Traps

Pitfall: Wrong inversion formula

  • Fix: $OP \cdot OP’ = r^2$, not $OP + OP’ = r^2$

Pitfall: Confusing inversion and reflection

  • Fix: Inversion uses circle, reflection uses line

Pitfall: Wrong spiral similarity setup

  • Fix: Spiral similarity combines rotation and scaling

Pitfall: Forgetting about angle preservation

  • Fix: Both inversion and spiral similarity preserve angles

🎯 AMC-Style Worked Example

Problem: In the figure, two circles are tangent at point $A$. A line through $A$ intersects the circles at points $B$ and $C$. If the circles have radii 3 and 5, and $AB = 4$, find $AC$.

Solution: This problem can be solved using inversion, but let’s use a simpler approach.

Since the circles are tangent at $A$, the line through $A$ is tangent to both circles.

Using the Power of a Point theorem:

  • For the first circle: $AB^2 = 4^2 = 16$
  • For the second circle: $AC^2 = 16$ (same power)

Therefore, $AC = 4$.

Answer: $AC = 4$

🔗 Related Topics

💡 Quick Reference

Inversion Properties

  • Formula: $OP \cdot OP’ = r^2$
  • Angle preservation: Yes
  • Circle mapping: Circles to circles or lines
  • Applications: Tangent circle problems

Spiral Similarity Properties

  • Combination: Rotation + homothety
  • Angle preservation: Yes
  • Distance scaling: By constant factor
  • Applications: Similar figure problems

Common Applications

  • Inversion: Tangent circles, angle preservation
  • Spiral similarity: Similar figures, ratio problems
  • Advanced problems: Complex geometric configurations

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