Numbering (computability theory) - Revision history

en>Wcherowi: /* Definition and examples */ clarified partial modifies function

2014-06-24T17:21:00Z

Definition and examples: clarified partial modifies function

← Older revision		Revision as of 19:21, 24 June 2014
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	~~[[Image:Triangle ABC with bisector AD.svg\|thumb\|240px\|right\|In this diagram, BD:DC = AB:AC.]]~~		Andrew Simcox is the title his mothers and fathers gave him and [http://chungmuroresidence.com/xe/reservation_branch2/152663 free psychic reading] he totally enjoys this name. He is an order [http://www.Sirudang.com/siroo_Notice/2110 cheap psychic readings] clerk and it's some thing he truly appreciate. What me and my family adore is to climb but I'm thinking on beginning some thing new. My spouse and I reside in Mississippi and I adore each working day living right here.<br><br>My blog post: clairvoyant psychic ([http://isaworld.pe.kr/?document_srl=392088 just click the following web page])
	~~In [[geometry]], the '''angle bisector theorem'''~~ is ~~concerned with~~ the ~~relative~~ [[length]]s of the two segments that a [[triangle]]'s side is divided into by a line that [[Bisection\|bisects]] the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

	Consider a triangle ''ABC''. Let the [[Bisection#Angle bisector\|angle bisector]] of angle ''A'' [[Line-line intersection\|intersect]] side ''BC'' at a point ''D''. The angle bisector theorem states that the ratio of the length of the [[line segment]] ''BD'' to the length of segment ''DC'' is equal to the ratio of the length of side ''AB'' to the length of side ''AC'':
	~~:<math>{\frac {\|BD\|} {\|DC\|}}={\frac {\|AB\|}{\|AC\|}}~~. </~~math>~~

	~~The generalized angle bisector theorem states that if D lies on BC, then~~

	~~:<math>{\frac {\|BD\|} {\|DC\|}}={\frac {\|AB\| \sin \angle DAB}{\|AC\| \sin \angle DAC}}. <~~/~~math>~~

	~~This reduces to the previous version if ''AD'' is the bisector of ''BAC''~~.

	~~The angle bisector theorem~~ is ~~commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.~~

	~~An angle bisector of~~ an ~~isosceles triangle will also bisect the opposite side, when the angle bisect bisects the vertex angle of the triangle~~

	~~== Proof ==~~
	~~In the above diagram, use the~~ [~~[law of sines]] on triangles ''ABD'' and ''ACD''~~:

	~~:<math>{\frac {\|AB\|} {\|BD\|}} = {\frac {\sin \angle BDA} {\sin \angle BAD}} <~~/~~math>~~ ..~~... (Equation 1)~~

	~~:<math>{\frac {\|AC\|} {\|DC\|}} = {\frac {\sin \angle ADC} {\sin \angle DAC}} <~~/~~math> ..... (Equation 2)~~

	~~Angles ''BDA'' and ''ADC'' form a linear pair, that is, they are adjacent [[supplementary angles]~~]~~. Since supplementary angles have equal sines,~~

	~~:<math>{{\sin \angle BDA}} = {\sin \angle ADC} </math>~~

	~~Angles ''BAD''~~ and '~~'DAC'' are equal~~. ~~Therefore the Right Hand Sides of Equations 1~~ and ~~2 are equal, so their Left Hand Sides must also be equal:~~

	~~:<math>{\frac {\|AB\|} {\|BD\|}}={\frac {\|AC\|}{\|DC\|}} </math>~~

	~~which~~ is ~~the Angle Bisector Theorem.~~

	~~If angles~~ '~~'BAD'' and ''DAC'' are unequal, Equations 1 and 2 can be re-written as:~~

	~~:<math> {\frac {\|AB\|} {\|BD\|} \sin \angle\ BAD = \sin \angle BDA}</math>~~

	~~:<math> {\frac {\|AC\|} {\|DC\|} \sin \angle\ DAC = \sin \angle ADC}~~.~~</math>~~

	~~Angles ''BDA''~~ and ~~''ADC'' are still supplementary, so the Right Hand Sides of these equations are equal, so the Left Hand Sides are equal to:~~

	~~:<math> {\frac {\|AB\|} {\|BD\|} \sin \angle\ BAD = \frac {\|AC\|} {\|DC\|} \sin \angle\ DAC}</math>~~

	~~which rearranges to the "generalized" version of the theorem.~~

	~~An alternative proof goes as follows, using its own diagram:~~

	~~[[Image:Bisekt.svg\|400px\|right]]~~

	~~Let ''B''<sub>1</sub> be the base of the altitude~~ in ~~the triangle ''ABD'' through ''B''~~ and ~~let ''C''<sub>1</sub> be the base of the altitude in the triangle ''ACD'' through ''C''~~. ~~Then,~~

	~~''DB''~~<~~sub~~>1<~~/sub~~>''B'' and ''DC''<sub>1</sub>''C'' are right, while the angles ''B''<sub>1</sub>''DB'' and ''C''<sub>1</sub>''DC'' are congruent if ''D'' lies on the segment ''BC'' and they are identical otherwise, so the triangles ''DB''<sub>1</sub>''B'' and ''DC''<sub>1</sub>''C'' are similar (~~AAA), which implies that~~

	~~:<math>{\frac {\|BD\|} {\|CD\|}}= {\frac {\|BB_1\|}{\|CC_1\|}}=\frac {\|AB\|\sin \angle BAD}{\|AC\|\sin \angle CAD}.</math>~~

	~~== External links ==~~
	* [http://~~www~~.~~cut-the-knot.org/Curriculum/Geometry/AngleBisectorRatio~~.~~shtml A Property of Angle Bisectors] at [[cut-the-knot]]~~
	* [http://planetmath.org/?op=~~getobj&from=objects&id=6487 Proof of angle bisector theorem~~] ~~at [[PlanetMath]]~~
	* [http://planetmath.org/?op=getobj&from=objects&id=6488 Another proof of angle bisector theorem] at [[PlanetMath]]
	* [http://gjarcmg.geometry-math-journal.ro/index/ On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem by G.W.I.S Amarasinghe, Global Journal of Advanced Research on Classical and Modern Geometries, Vol 01(01)~~, pp. 15 - 27, 2012]~~
	~~[[Category:Articles containing proofs]]~~
	~~[[Category:Elementary geometry]]~~
	~~[[Category:Triangle geometry]]~~
	~~[[Category:Theorems in plane geometry]]~~

77.253.78.118: /* Types of numberings */

2013-09-05T20:58:22Z

Types of numberings

← Older revision		Revision as of 22:58, 5 September 2013
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	~~Alyson~~ is the ~~title people use~~ to ~~call me~~ and ~~I think it sounds quite good~~ when ~~you say it~~. ~~To play lacross~~ is the ~~thing I love most~~ of ~~all~~. ~~My spouse~~ and ~~I reside~~ in ~~Mississippi~~ and ~~I love every working day residing~~ right ~~here. Office supervising is what she does for a residing.~~<br><br>~~Stop by my page~~ [http://~~ustanford~~.~~com~~/~~index~~.~~php~~?do=/~~profile~~-~~38218~~/~~info~~/ ~~telephone psychic~~]		[[Image:Triangle ABC with bisector AD.svg\|thumb\|240px\|right\|In this diagram, BD:DC = AB:AC.]]
			In [[geometry]], the '''angle bisector theorem''' is concerned with the relative [[length]]s of the two segments that a [[triangle]]'s side is divided into by a line that [[Bisection\|bisects]] the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

			Consider a triangle ''ABC''. Let the [[Bisection#Angle bisector\|angle bisector]] of angle ''A'' [[Line-line intersection\|intersect]] side ''BC'' at a point ''D''. The angle bisector theorem states that the ratio of the length of the [[line segment]] ''BD'' to the length of segment ''DC'' is equal to the ratio of the length of side ''AB'' to the length of side ''AC'':
			:<math>{\frac {\|BD\|} {\|DC\|}}={\frac {\|AB\|}{\|AC\|}}. </math>

			The generalized angle bisector theorem states that if D lies on BC, then

			:<math>{\frac {\|BD\|} {\|DC\|}}={\frac {\|AB\| \sin \angle DAB}{\|AC\| \sin \angle DAC}}. </math>

			This reduces to the previous version if ''AD'' is the bisector of ''BAC''.

			The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.

			An angle bisector of an isosceles triangle will also bisect the opposite side, when the angle bisect bisects the vertex angle of the triangle

			== Proof ==
			In the above diagram, use the [[law of sines]] on triangles ''ABD'' and ''ACD'':

			:<math>{\frac {\|AB\|} {\|BD\|}} = {\frac {\sin \angle BDA} {\sin \angle BAD}} </math> ..... (Equation 1)

			:<math>{\frac {\|AC\|} {\|DC\|}} = {\frac {\sin \angle ADC} {\sin \angle DAC}} </math> ..... (Equation 2)

			Angles ''BDA'' and ''ADC'' form a linear pair, that is, they are adjacent [[supplementary angles]]. Since supplementary angles have equal sines,

			:<math>{{\sin \angle BDA}} = {\sin \angle ADC} </math>

			Angles ''BAD'' and ''DAC'' are equal. Therefore the Right Hand Sides of Equations 1 and 2 are equal, so their Left Hand Sides must also be equal:

			:<math>{\frac {\|AB\|} {\|BD\|}}={\frac {\|AC\|}{\|DC\|}} </math>

			which is the Angle Bisector Theorem.

			If angles ''BAD'' and ''DAC'' are unequal, Equations 1 and 2 can be re-written as:

			:<math> {\frac {\|AB\|} {\|BD\|} \sin \angle\ BAD = \sin \angle BDA}</math>

			:<math> {\frac {\|AC\|} {\|DC\|} \sin \angle\ DAC = \sin \angle ADC}.</math>

			Angles ''BDA'' and ''ADC'' are still supplementary, so the Right Hand Sides of these equations are equal, so the Left Hand Sides are equal to:

			:<math> {\frac {\|AB\|} {\|BD\|} \sin \angle\ BAD = \frac {\|AC\|} {\|DC\|} \sin \angle\ DAC}</math>

			which rearranges to the "generalized" version of the theorem.

			An alternative proof goes as follows, using its own diagram:

			[[Image:Bisekt.svg\|400px\|right]]

			Let ''B''<sub>1</sub> be the base of the altitude in the triangle ''ABD'' through ''B'' and let ''C''<sub>1</sub> be the base of the altitude in the triangle ''ACD'' through ''C''. Then,

			''DB''<sub>1</sub>''B'' and ''DC''<sub>1</sub>''C'' are right, while the angles ''B''<sub>1</sub>''DB'' and ''C''<sub>1</sub>''DC'' are congruent if ''D'' lies on the segment ''BC'' and they are identical otherwise, so the triangles ''DB''<sub>1</sub>''B'' and ''DC''<sub>1</sub>''C'' are similar (AAA), which implies that

			:<math>{\frac {\|BD\|} {\|CD\|}}= {\frac {\|BB_1\|}{\|CC_1\|}}=\frac {\|AB\|\sin \angle BAD}{\|AC\|\sin \angle CAD}.</math>

			== External links ==
			* [http://www.cut-the-knot.org/Curriculum/Geometry/AngleBisectorRatio.shtml A Property of Angle Bisectors] at [[cut-the-knot]]
			* [http://planetmath.org/?op=getobj&from=objects&id=6487 Proof of angle bisector theorem] at [[PlanetMath]]
			* [http://planetmath.org/?op=getobj&from=objects&id=6488 Another proof of angle bisector theorem] at [[PlanetMath]]
			* [http://gjarcmg.geometry-math-journal.ro/index/ On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem by G.W.I.S Amarasinghe, Global Journal of Advanced Research on Classical and Modern Geometries, Vol 01(01), pp. 15 - 27, 2012]
			[[Category:Articles containing proofs]]
			[[Category:Elementary geometry]]
			[[Category:Triangle geometry]]
			[[Category:Theorems in plane geometry]]

en>Arrr2 at 18:05, 27 December 2011

2011-12-27T18:05:40Z

New page

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