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.
Content ©2008. All Rights Reserved.
Date last modified: May 24, 2009.