Inscribing a quadrilateral in a circle

[ehrenfest]

Homework Statement

If both pairs of opposite angles of a quadrilateral add up to 180, is it always possible to inscribe it in a circle?

Homework Equations

The Attempt at a Solution

The converse is easily proven, since the angle between two chords standing on the circumfrence is half of the corresponding central angle.

 

[chaoseverlasting]

If by inscribe, you mean that all four vertices lie on the circle, I don't think so.

 

[Dick]

If by inscribe, you mean that all four vertices lie on the circle, I don't think so.

It is true. Draw the circle containing three points of the quadrilateral and then use the supplementary angle property to show the fourth point must lie on the circle.

 

[chaoseverlasting]

Yes, but if only two points lie on the circle, then the other two don't necessarily have to. Thats what I was thinking about. The same obviously goes for one point lying on the circle.

In any case, the opposite angles of a quadrilateral will always be supplementary.

 

[ehrenfest]

In any case, the opposite angles of a quadrilateral will always be supplementary.

No, e.g. a diamond.

 

[Dick]

Yes, but if only two points lie on the circle, then the other two don't necessarily have to. Thats what I was thinking about. The same obviously goes for one point lying on the circle.

In any case, the opposite angles of a quadrilateral will always be supplementary.

You can put a circle through any three noncollinear points.