Difference between revisions of "Cyclic quadrilateral"

(Properties)
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* <math>\angle CAD = \angle CBD</math>
 
* <math>\angle CAD = \angle CBD</math>
 
* All four [[perpendicular bisector|perpendicular bisectors]] are [[concurrent]]. The converse is also true. This intersection is the [[circumcenter]] of the quadrilateral.
 
* All four [[perpendicular bisector|perpendicular bisectors]] are [[concurrent]]. The converse is also true. This intersection is the [[circumcenter]] of the quadrilateral.
* Any two opposite sites of the quadrilateral are antiparallel with respect to the other two opposite sites.
+
* Any two opposite sites of the quadrilateral are antiparallel with respect [https://artofproblemsolving.com/wiki/index.php/TOTO_SLOT_:_SITUS_TOTO_SLOT_MAXWIN_TERBAIK_DAN_TERPERCAYA TOTO SLOT] to the other two opposite sites.
 
*ef=ac+bd (e and f are the diagonals)
 
*ef=ac+bd (e and f are the diagonals)
  

Revision as of 15:59, 19 February 2024

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. While all triangles are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties.

Cyclicquad2.png

Properties

In a quadrilateral $ABCD$:

  • $\angle A + \angle C = \angle B + \angle D = {180}^{o}$ This property is both sufficient and necessary, and is often used to show that a quadrilateral is cyclic.
  • $\angle ABD = \angle ACD$
  • $\angle BCA = \angle BDA$
  • $\angle BAC = \angle BDC$
  • $\angle CAD = \angle CBD$
  • All four perpendicular bisectors are concurrent. The converse is also true. This intersection is the circumcenter of the quadrilateral.
  • Any two opposite sites of the quadrilateral are antiparallel with respect TOTO SLOT to the other two opposite sites.
  • ef=ac+bd (e and f are the diagonals)

Applicable Theorems/Formulae

The following theorems and formulae apply to cyclic quadrilaterals:

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