In convex quadrilateral $ABCD$, angles $ABC$ and $ADC$ are supplementary; angles $BAD$ and \

Question

In convex 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)?

Accepted Answer

The supplementary opposite-angle condition proves $A,B,C,D$ concyclic. Their chord perpendicular bisectors meet at the circumcentre. See the solution for the incircle criterion.

Proofs of Conditions of a Cyclic Quadrilateral

Proofs of Conditions of a Cyclic Quadrilateral

Solution:

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