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| [[Image:Triangle ABC with bisector AD.svg|thumb|240px|right|In this diagram, BD:DC = AB:AC.]]
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| 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.
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| 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'':
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| :<math>{\frac {|BD|} {|DC|}}={\frac {|AB|}{|AC|}}. </math>
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| The generalized angle bisector theorem states that if D lies on BC, then
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| :<math>{\frac {|BD|} {|DC|}}={\frac {|AB| \sin \angle DAB}{|AC| \sin \angle DAC}}. </math>
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| This reduces to the previous version if ''AD'' is the bisector of ''BAC''.
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| 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.
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| An angle bisector of an isosceles triangle will also bisect the opposite side, when the angle bisect bisects the vertex angle of the triangle
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| == Proof ==
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| In the above diagram, use the [[law of sines]] on triangles ''ABD'' and ''ACD'':
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| :<math>{\frac {|AB|} {|BD|}} = {\frac {\sin \angle BDA} {\sin \angle BAD}} </math> ..... (Equation 1)
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| :<math>{\frac {|AC|} {|DC|}} = {\frac {\sin \angle ADC} {\sin \angle DAC}} </math> ..... (Equation 2)
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| Angles ''BDA'' and ''ADC'' form a linear pair, that is, they are adjacent [[supplementary angles]]. Since supplementary angles have equal sines,
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| :<math>{{\sin \angle BDA}} = {\sin \angle ADC} </math>
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| 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:
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| :<math>{\frac {|AB|} {|BD|}}={\frac {|AC|}{|DC|}} </math>
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| which is the Angle Bisector Theorem.
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| If angles ''BAD'' and ''DAC'' are unequal, Equations 1 and 2 can be re-written as:
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| :<math> {\frac {|AB|} {|BD|} \sin \angle\ BAD = \sin \angle BDA}</math>
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| :<math> {\frac {|AC|} {|DC|} \sin \angle\ DAC = \sin \angle ADC}.</math>
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| 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:
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| :<math> {\frac {|AB|} {|BD|} \sin \angle\ BAD = \frac {|AC|} {|DC|} \sin \angle\ DAC}</math>
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| which rearranges to the "generalized" version of the theorem.
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| An alternative proof goes as follows, using its own diagram:
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| [[Image:Bisekt.svg|400px|right]]
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| 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,
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| ''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
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| :<math>{\frac {|BD|} {|CD|}}= {\frac {|BB_1|}{|CC_1|}}=\frac {|AB|\sin \angle BAD}{|AC|\sin \angle CAD}.</math>
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| == External links ==
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/AngleBisectorRatio.shtml A Property of Angle Bisectors] at [[cut-the-knot]]
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| * [http://planetmath.org/?op=getobj&from=objects&id=6487 Proof of angle bisector theorem] at [[PlanetMath]]
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| * [http://planetmath.org/?op=getobj&from=objects&id=6488 Another proof of angle bisector theorem] at [[PlanetMath]]
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| * [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]
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| [[Category:Articles containing proofs]]
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| [[Category:Elementary geometry]]
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| [[Category:Triangle geometry]]
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| [[Category:Theorems in plane geometry]]
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