🔗 Tangency Configurations

Tangency problems are common in AMC geometry, involving incircles, excircles, and tangent chains. Master these configurations for contest success.

🎯 Key Concepts

Incircles

Definition: Circle inscribed in triangle, tangent to all three sides Properties:

  • Center is the incenter (intersection of angle bisectors)
  • Radius is the inradius
  • Area formula: $A = rs$ where $r$ is inradius, $s$ is semi-perimeter

Excircles

Definition: Circle tangent to one side and extensions of other two sides Properties:

  • Center is the excenter (intersection of external angle bisectors)
  • Radius is the exradius
  • Three excircles, one opposite each vertex

Tangent Chains

Definition: Series of circles each tangent to the next Properties:

  • Equal tangent lengths from external point
  • Power of a point applications
  • Similarity relationships

Equal Tangents

Key Property: From external point, two tangents to circle are equal Applications:

  • Finding equal lengths
  • Setting up equations
  • Proving equalities

🔍 Micro-Examples

Incircle Example

Triangle with sides 3, 4, 5:

  • Semi-perimeter: $s = \frac{3+4+5}{2} = 6$
  • Area: $A = 6$ (using Heron’s formula)
  • Inradius: $r = \frac{A}{s} = \frac{6}{6} = 1$

Excircle Example

Triangle with sides 3, 4, 5, excircle opposite side 3:

  • Semi-perimeter: $s = 6$
  • Area: $A = 6$
  • Exradius: $r_a = \frac{A}{s-a} = \frac{6}{6-3} = 2$

Equal Tangents Example

From external point, two tangents to circle of radius 5:

  • If one tangent is 12, the other is also 12
  • Distance from point to center: $\sqrt{12^2 + 5^2} = 13$

⚠️ Common Traps

Pitfall: Confusing incircle and excircle

  • Fix: Incircle is inside, excircle is outside

Pitfall: Wrong inradius formula

  • Fix: $r = \frac{A}{s}$, not $\frac{A}{P}$

Pitfall: Forgetting about equal tangents

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

Pitfall: Wrong tangent chain setup

  • Fix: Use Power of a Point for tangent chains

🎯 AMC-Style Worked Example

Problem: In triangle $ABC$, the incircle is tangent to sides $AB$, $BC$, $CA$ at points $D$, $E$, $F$ respectively. If $AB = 5$, $BC = 6$, $CA = 7$, find the length of $AD$.

Solution: First, find the semi-perimeter: $s = \frac{5+6+7}{2} = 9$

Using the property that tangents from external point are equal:

  • $AD = AF$ (tangents from $A$)
  • $BD = BE$ (tangents from $B$)
  • $CE = CF$ (tangents from $C$)

Let $AD = AF = x$, $BD = BE = y$, $CE = CF = z$.

From the side lengths:

  • $x + y = 5$ (side $AB$)
  • $y + z = 6$ (side $BC$)
  • $z + x = 7$ (side $CA$)

Adding all three equations: $2(x + y + z) = 18$ $x + y + z = 9$

Since $x + y = 5$, we have $z = 4$. Since $z + x = 7$, we have $x = 3$.

Answer: $AD = 3$

🔗 Related Topics

💡 Quick Reference

Incircle Properties

  • Center: Incenter (intersection of angle bisectors)
  • Radius: $r = \frac{A}{s}$ (area divided by semi-perimeter)
  • Tangency: Tangent to all three sides

Excircle Properties

  • Center: Excenter (intersection of external angle bisectors)
  • Radius: $r_a = \frac{A}{s-a}$ (area divided by semi-perimeter minus opposite side)
  • Tangency: Tangent to one side and extensions of other two

Equal Tangents

  • Property: From external point, two tangents are equal
  • Applications: Finding equal lengths, setting up equations
  • Power of a Point: Use for tangent-secant relationships

Common Applications

  • Incircles: Finding inradius, tangent lengths
  • Excircles: Finding exradius, tangent lengths
  • Tangent chains: Using Power of a Point
  • Equal tangents: Proving equalities

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