🔄 Cyclic Quadrilaterals

A cyclic quadrilateral is one that can be inscribed in a circle. These quadrilaterals have special properties that make them powerful tools in AMC geometry.

🎯 Key Concepts

Definition and Recognition

A quadrilateral is cyclic if and only if all four vertices lie on a circle.

Recognition Criteria:

  • Opposite angles sum to $180°$
  • Equal angles subtend the same arc
  • One angle equals the opposite angle’s supplement

Opposite Angle Theorem

Fundamental: In a cyclic quadrilateral, opposite angles are supplementary.

Formula: $\angle A + \angle C = 180°$ and $\angle B + \angle D = 180°$

Ptolemy’s Theorem

Powerful Tool: In a cyclic quadrilateral $ABCD$: $AC \cdot BD = AB \cdot CD + BC \cdot AD$

Special Case: For a rectangle (which is cyclic), this becomes the Pythagorean theorem.

Equal Subtended Angles

Key Insight: Angles subtending the same arc are equal.

Applications:

  • Proving quadrilaterals are cyclic
  • Finding equal angles in complex figures
  • Setting up similarity relationships

🔍 Micro-Examples

Opposite Angles

In cyclic quadrilateral $ABCD$, if $\angle A = 70°$, then $\angle C = 180° - 70° = 110°$.

Ptolemy’s Theorem

In cyclic quadrilateral with sides 3, 4, 5, 6, if diagonals are $d_1$ and $d_2$, then $d_1 \cdot d_2 = 3 \cdot 5 + 4 \cdot 6 = 15 + 24 = 39$.

Equal Subtended Angles

If $\angle ACB = \angle ADB$, then quadrilateral $ABCD$ is cyclic.

⚠️ Common Traps

Pitfall: Assuming quadrilateral is cyclic without proof

  • Fix: Use opposite angle test or equal subtended angles

Pitfall: Wrong order in Ptolemy’s theorem

  • 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

🎯 AMC-Style Worked Example

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

Solution: Using Ptolemy’s theorem for cyclic quadrilateral $ABCD$:

$AC \cdot BD = AB \cdot CD + BC \cdot AD$

Substituting the given values: $7 \cdot BD = 3 \cdot 5 + 4 \cdot 6$ $7 \cdot BD = 15 + 24$ $7 \cdot BD = 39$ $BD = \frac{39}{7}$

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

🔗 Related Topics

💡 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

Special Cases

  • Rectangle: Always cyclic, diagonals are diameters
  • Square: Always cyclic, diagonals are equal and perpendicular
  • Isosceles trapezoid: Always cyclic if non-parallel sides are equal

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