In quadrilateral \(ABCD\), angles \(ABC\) and \(ADC\) are supplementary; angles \(BAD\) and \(BCD\) are supplementary. Prove that there exists a circle with centre \(O\) that passes through \(A\), \(B\), \(C\) and \(D\).
Hint: Prove by contradiction. Suppose on the contrary, there exists no such circle that passes through \(A\), \(B\), \(C\) and \(D\).
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Note: In this case, the \(4\) perpendicular bisectors of the \(4\) sides of quadrilateral \(ABCD\) are concurrent, where&nbsp;\(O\) is the common point. However, this does not hold in general.
Think: What condition(s) must a quadrilateral satisfy such that there exists a circle that touches the \(4\) sides of the quadrilateral at one point, respectively (inscribed circle)?

In quadrilateral \(ABCD\), angles \(ABC\) and \(ADC\) are supplementary; angles \(BAD\) and \(BCD\) are supplementary. Prove that there exists a circle with centre \(O\) that passes through \(A\), \(B\), \(C\) and \(D\).

Hint: Prove by contradiction. Suppose on the contrary, there exists no such circle that passes through \(A\), \(B\), \(C\) and \(D\).

**Note:** In this case, the \(4\) perpendicular bisectors of the \(4\) sides of quadrilateral \(ABCD\) are concurrent, where \(O\) is the common point. However, this does not hold in general.

**Think:** What condition(s) must a quadrilateral satisfy such that there exists a circle that touches the \(4\) sides of the quadrilateral at one point, respectively (inscribed circle)?

Proofs of Conditions of a Cyclic Quadrilateral

Proofs of Conditions of a Cyclic Quadrilateral