வட்ட நாற்கரம்: திருத்தங்களுக்கு இடையிலான வேறுபாடு
உள்ளடக்கம் நீக்கப்பட்டது உள்ளடக்கம் சேர்க்கப்பட்டது
வரிசை 4:
==சிறப்பு வகைகள்==
[[சதுரம்]], [[செவ்வகம்]], [[இருசமபக்க சரிவகம்]], [[எதிர் இணைகரம்]] ஆகியவை அனைத்தும் வட்ட நாற்கரங்களாகவே அமையும். ஒரு [[பட்டம் (வடிவவியல்)|பட்ட நாற்கரத்தின்]] இரு கோணங்கள் [[செங்கோணம்|செங்கோணங்களாக]] இருந்தால், இருந்தால் மட்டுமே அது வட்ட நாற்கரமாக இருக்க முடியும். [[இரு மைய நாற்கரம்]] வட்ட நாற்கரமாகவும் [[தொடு நாற்கரம்|தொடு நாற்கரமாகவும்]] இருக்கும். அதேபோல [[வெளி-தொடு நாற்கரம்#வெளி-இரு மைய நாற்கரம்|வெளி-இரு மைய நாற்கரமும்]] வட்ட நாற்கரமாகவும் [[வெளி-தொடு நாற்கரம்|வெளி-தொடு நாற்கரமாகவும்]] இருக்கும்.
==பண்புகள்==
A convex quadrilateral is cyclic [[if and only if]] the four [[perpendicular]] bisectors to the sides are [[concurrent]]. This common point is the [[circumcenter]].<ref name=Usiskin>{{citation|last1=Usiskin|first1=Zalman|last2=Griffin|first2=Jennifer|title=The Classification of Quadrilaterals. A Study of Definition|publisher=Information Age Publishing|year=2008|pages=63–65|isbn=1593116950}}.</ref>
A convex quadrilateral ''ABCD'' is cyclic if and only if its opposite angles are [[supplementary angle|supplementary]], that is<ref name=Usiskin/>
:<math>A + C = B + D = \pi = 180^{\circ}.</math>
The direct theorem was Proposition 22 in Book 3 of [[Euclid]]'s [[Euclid's Elements|''Elements'']].<ref>[http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII22.html Book 3, Proposition 22] of [[Euclid's Elements]].</ref> Equivalently, a convex quadrilateral is cyclic if and only if each [[exterior angle]] is equal to the opposite [[interior angle]].
Another [[necessary and sufficient condition]] for a convex quadrilateral ''ABCD'' to be cyclic is that an angle between a side and a [[diagonal]] is equal to the angle between the opposite side and the other diagonal.<ref name=Andreescu>{{citation
| last1 = Andreescu | first1 = Titu
| last2 = Enescu | first2 = Bogdan
| isbn = 0-8176-4305-2
| location = Boston, MA
| mr = 2025063
| pages = 44–46, 50
| publisher = Birkhäuser Boston Inc.
| title = Mathematical olympiad treasures
| year = 2004}}.</ref> That is, for example,
:<math>\angle ACB = \angle ADB.</math>
[[Ptolemy's theorem]] expresses the product of the lengths of the two diagonals ''p'' and ''q'' of a cyclic quadrilateral as equal to the sum of the products of opposite sides:<ref name=Durell/>{{rp|p.25}}
:<math>\displaystyle pq = ac + bd.</math>
The [[Theorem#Converse|converse]] is also true. That is, if this equation is satisfied in a convex quadrilateral, then it is a cyclic quadrilateral.
If two chords of a circle ''AC'' and ''BD'' intersect at ''X'', then the four points ''A'', ''B'', ''C'', ''D'' are concyclic if and only if<ref>Bradley, Christopher J., ''The algebra of geometry. Cartesian, Areal and Projective co-ordinates'', Highperception, 2007, p. 179.</ref>
:<math>\displaystyle AX\cdot XC = BX\cdot XD.</math>
The intersection ''X'' may be internal or external to the circle. In the former case, the quadrilateral is ''ABCD'', and in the latter case, the convex quadrilateral is ''ABDC''. When the intersection is internal, the equality states that the product of the segment lengths into which ''X'' divides one diagonal equals that of the other diagonal. This is known as the ''intersecting chords theorem'' since the diagonals of the cyclic quadrilateral are chords of the circumcircle.
Yet another characterization is that a convex quadrilateral ''ABCD'' is cyclic if and only if<ref>{{citation
|last=Hajja |first=Mowaffaq
|journal=Forum Geometricorum
|pages=103–106
|title=A condition for a circumscriptible quadrilateral to be cyclic
|url=http://forumgeom.fau.edu/FG2008volume8/FG200814.pdf
|volume=8
|year=2008}}.</ref>
:<math>\tan{\frac{A}{2}}\tan{\frac{C}{2}}=\tan{\frac{B}{2}}\tan{\frac{D}{2}}=1.</math>
==பிரம்மகுப்தரின் நாற்கரம்==
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