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Toolkit 61

4 Rules of Cyclic Quadrilaterals

1. Opposite angles are supplementary

Cyclic quadrilateral ABCD inscribed in a circle

$ABCD$ is cyclic if and only if:

\angle A+\angle C=180^\circ

or

\angle B+\angle D=180^\circ

2. Equal angles subtend the same chord

Cyclic quadrilateral with equal angles at A and C

$ABCD$ is cyclic if and only if:

\angle A_1=\angle C_1

3. Intersecting chords

Diagonals of cyclic quadrilateral intersecting at P inside the circle

$ABCD$ is cyclic if and only if:

PA\cdot PC=PB\cdot PD

4. Power of a point

Secants from external point P meeting the circle

$ABCD$ is cyclic if and only if:

PA\cdot PD=PB\cdot PC

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