Cyclic quadrilaterals

A cyclic Quadrilateral is a special type of quadrilateral that can be inscribed inside of a Circle.

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Properties

• $\angle {ABC}+\angle {ADC}=\angle {BCD}+\angle {BAD}=180^{\circ }$.
• $\angle {ABD}=\angle {ACD}$.
• $\angle {ACB}=\angle {ADB}$.
• $\angle {BAC}=\angle {BDC}$.
• $\angle {CBD}=\angle {CAD}$.

Common Cyclic Quadrilaterals

• Square
• Parallelogram
• Isosceles trapezoid
• Rectangle

Ptolemy's Theorem

Ptolemy's Theorem states that

$(AB)(CD)+(BC)(AD)=(AC)(BD).$ This theorem can be proved by drawing point

$E$ on

${\overline {BD}}$ such that

$\Delta ADE\sim \Delta ABC.$ From triangle similarity, we have that

${\frac {BC}{AC}}={\frac {ED}{AD}}.$ and

${\frac {AB}{BE}}={\frac {AC}{CD}}.$ Cross-multiplying and then adding both equations gives us

$(AB)(CD)+(BC)(AD)=(AC)(ED+BE).$ We know that

$BD=ED+BE$ , so hence,

$(AB)(CD)+(BC)(AD)=(AC)(BD).$