In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

Theorem

Consider a triangle △ABC. Let the angle bisector of angle ∠ A intersect side BC at a point D between B and C. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC:

${\displaystyle {\frac {|BD|}{|CD|}}={\frac {|AB|}{|AC|}},}$

and conversely, if a point D on the side BC of △ABC divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle ∠ A.

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

${\displaystyle {\frac {|BD|}{|CD|}}={\frac {|AB|\sin \angle DAB}{|AC|\sin \angle DAC}}.}$

This reduces to the previous version if AD is the bisector of ∠ BAC. When D is external to the segment BC, directed line segments and directed angles must be used in the calculation.

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 immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.

Proofs

There exist many different ways of proving the angle bisector theorem. A few of them are shown below.

Proof using similar triangles

As shown in the accompanying animation, the theorem can be proved using similar triangles. In the version illustrated here, the triangle ${\displaystyle \triangle ABC}$ gets reflected across a line that is perpendicular to the angle bisector ${\displaystyle AD}$, resulting in the triangle ${\displaystyle \triangle AB_{2}C_{2}}$ with bisector ${\displaystyle AD_{2}}$. The fact that the bisection-produced angles ${\displaystyle \angle BAD}$ and ${\displaystyle \angle CAD}$ are equal means that ${\displaystyle BAC_{2}}$ and ${\displaystyle CAB_{2}}$ are straight lines. This allows the construction of triangle ${\displaystyle \triangle C_{2}BC}$ that is similar to ${\displaystyle \triangle ABD}$. Because the ratios between corresponding sides of similar triangles are all equal, it follows that ${\displaystyle |AB|/|AC_{2}|=|BD|/|CD|}$. However, ${\displaystyle AC_{2}}$ was constructed as a reflection of the line ${\displaystyle AC}$, and so those two lines are of equal length. Therefore, ${\displaystyle |AB|/|AC|=|BD|/|CD|}$, yielding the result stated by the theorem.

Proof using Law of Sines

In the above diagram, use the law of sines on triangles △ABD and △ACD:

${\displaystyle {\frac {|AB|}{|BD|}}={\frac {\sin \angle ADB}{\sin \angle DAB}}}$

(1)

${\displaystyle {\frac {|AC|}{|CD|}}={\frac {\sin \angle ADC}{\sin \angle DAC}}}$

(2)

Angles ∠ ADB and ∠ ADC form a linear pair, that is, they are adjacent supplementary angles. Since supplementary angles have equal sines,

${\displaystyle {\sin \angle ADB}={\sin \angle ADC}.}$

Angles ∠ DAB 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.

${\displaystyle {\frac {|BD|}{|CD|}}={\frac {|AB|}{|AC|}},}$

which is the angle bisector theorem.

If angles ∠ DAB, ∠ DAC are unequal, equations (1) and (2) can be re-written as:

${\displaystyle \frac{|AB|}{|BD|}\sin <!--\; \; -->\mathrm{\angle <!--\; \angle \; -->}DAB=\sin <!--\; \; -->}$Zdroj:https://en.wikipedia.org?pojem=Angle_bisector_theorem
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