Overview

Cyclic quadrilaterals are defined by opposite angles summing to $180^\circ$. They unlock Ptolemy and Brahmagupta formulas.

Key Ideas

• $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
• Ptolemy: $AC\cdot BD = AB\cdot CD + AD\cdot BC$.

Core Skills

Prove Cyclicity

Show a pair of opposite angles sum to $180^\circ$, or show equal angles subtend the same chord.

Use Ptolemy for Lengths

If all four sides and one diagonal are known, Ptolemy gives the other diagonal.

Angle Chasing with Arcs

Inscribed angles that intercept the same arc are equal; use this to move angles around the quadrilateral.

Worked Example

Cyclic quadrilateral has $\angle A = 70^\circ$. Find $\angle C$.

$\angle C = 180^\circ - 70^\circ = 110^\circ$.

More Examples

Example 1: Ptolemy

In cyclic $ABCD$, $AB=3$, $BC=4$, $CD=5$, $AD=6$, and diagonal $AC=7$. Find $BD$.

$7\cdot BD = 3\cdot 5 + 6\cdot 4 = 39$, so $BD=39/7$.

Example 2: Cyclic Test

If $\angle A = 85^\circ$ and $\angle C=95^\circ$, is $ABCD$ cyclic?

Yes, since $\angle A + \angle C = 180^\circ$.

Strategy Checklist

• Confirm cyclicity before using Ptolemy.
• Use opposite-angle sums for quick angle results.
• Track arcs when moving angles across the circle.

Common Pitfalls

• Assuming a quadrilateral is cyclic without proof.
• Using Ptolemy on non-cyclic quadrilaterals.
• Mixing up diagonals in Ptolemy's formula.

Practice Problems