Inscribed Quadrilaterals

As we mentioned before, not all quadrilaterals can be inscribed in a circle.  A quadrilateral that can be inscribed in a circle is called a cyclic quadrilateral.  The main result about cyclic quadrilaterals is:

Opposite angels of a cyclic quadrilateral are supplementary, and conversely.

To see why this must be so, suppose ABCD is a cyclic quadrilateral inscribed in circle O:

Then angle A measures half of arc BCD and angle C measures half of arc DAB since these are inscribed angles.  But arcs BCD and DAB add to 360o, so angles A and C add to 180o.  Likewise, angles B and D add to 180o, which proves opposite angles of a cyclic quadrilateral are supplementary.

The converse is also true:

Suppose opposite angles of quadrilateral ABCD are supplementary.  That is, angles A and C are supplementary, and angles B and D are supplementary:  Triangle BCD is cyclic, so construct the circle which contains it.  Suppose A does not lie on that circle:

Then let P be the point of intersection of AB with that circle, and draw segment DP:

Since quadrilateral PBCD is cyclic, angle DPB is supplementary to angle C.  But since angle A is also supplementary to angle C, angles DPB and A are congruent.  But angle DPB is an exterior angle of triangle APD, so it would have to be greater than the interior angle A.  This is a contradiction, so it must be the case that A does not lie outside the circle.  By a similar argument, it is easy to show that A cannot lie inside that circle either, so it must be that A lies on that circle, and therefore ABCD is cyclic.