🔄 Cyclic Quad Setups

Cyclic quadrilaterals have special properties that make them powerful tools in AMC geometry. Master these patterns for efficient problem solving.

🎯 Recognition Cues

Key Triggers

  • Equal subtended angles - Look for angles that subtend the same arc
  • Opposite angle sums - Look for angles that sum to $180°$
  • Ptolemy’s theorem - Look for products of opposite sides
  • Circle properties - Look for quadrilaterals inscribed in circles

Common Setups

  • Quadrilaterals with equal subtended angles
  • Opposite angles that sum to $180°$
  • Products of opposite sides
  • Inscribed quadrilaterals

🧩 Solution Template

Step 1: Identify Cyclicity

  • Check if opposite angles sum to $180°$
  • Look for equal subtended angles
  • Verify that all four vertices lie on a circle

Step 2: Apply Properties

  • Opposite angles: Sum to $180°$
  • Equal subtended angles: Angles subtending same arc are equal
  • Ptolemy’s theorem: $AC \cdot BD = AB \cdot CD + BC \cdot AD$

Step 3: Set Up Equations

  • Use the appropriate property
  • Substitute known values
  • Solve for the unknown

Step 4: Verify

  • Check that the quadrilateral is actually cyclic
  • Ensure all properties are satisfied
  • Verify the final answer

🔍 Worked Example

Problem: In cyclic quadrilateral $ABCD$, $AB = 3$, $BC = 4$, $CD = 5$, and $DA = 6$. If $AC = 7$, find $BD$.

Solution: Step 1: Identify cyclicity

  • Given that $ABCD$ is cyclic
  • Can use Ptolemy’s theorem

Step 2: Apply properties

  • Use Ptolemy’s theorem: $AC \cdot BD = AB \cdot CD + BC \cdot AD$

Step 3: Set up equation

  • $AC \cdot BD = AB \cdot CD + BC \cdot AD$
  • $7 \cdot BD = 3 \cdot 5 + 4 \cdot 6$
  • $7 \cdot BD = 15 + 24$
  • $7 \cdot BD = 39$
  • $BD = \frac{39}{7}$

Step 4: Verify

  • Check that $ABCD$ is cyclic ✓
  • Ptolemy’s theorem applies ✓
  • Answer is positive ✓

Answer: $BD = \frac{39}{7}$

⚠️ Common Pitfalls

Pitfall: Assuming quadrilateral is cyclic without proof

  • Fix: Use opposite angle test or equal subtended angles

Pitfall: Wrong Ptolemy’s theorem setup

  • Fix: Remember $AC \cdot BD = AB \cdot CD + BC \cdot AD$

Pitfall: Confusing cyclic and circumscribed

  • Fix: Cyclic means inscribed in circle, circumscribed means circle inscribed in quadrilateral

Pitfall: Forgetting about equal subtended angles

  • Fix: Look for angles that subtend the same arc

🔗 Related Patterns

💡 Quick Reference

Cyclic Quadrilateral Properties

  • Opposite angles: Sum to $180°$
  • Equal subtended angles: Angles subtending same arc are equal
  • Ptolemy’s theorem: $AC \cdot BD = AB \cdot CD + BC \cdot AD$

Recognition Tests

  • Opposite angle test: Check if opposite angles sum to $180°$
  • Equal subtended angles: Look for angles subtending same arc
  • Power of a point: Use Power of a Point to prove cyclicity

Solution Strategy

  • Identify: Look for cyclic quadrilateral properties
  • Prove: Use recognition tests if not given
  • Apply: Use appropriate property (opposite angles, Ptolemy)
  • Calculate: Solve for unknown values

Next: Area Ratio in Triangle → | Prev: Tangent-Secant-Chord → | Back to: Geometry Mastery Guide →