martes, 30 de marzo de 2021

Length of angle bisector: yet another application of the half-angle formulas

Let $\triangle{ABC}$ be a triangle. Let $AD$ be the angle bisector of $\angle{BAC}$ in $\triangle{ABC}$. Let $d$ be the length of $AD$. Then $d$ is given by:

$$d^2=\frac{bc}{(b+c)^2}\left[(b+c)^2-a^2\right]\tag{1}$$

where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.

Lemma 1 (Tehebycheff). Let $\triangle{ABC}$ be a triangle. Let $AD$ be the angle bisector of $\angle{BAC}$ in $\triangle{ABC}$. Let $d$ be the length of $AD$ and $\angle{BAD}=\angle{CAD}=\frac{\alpha}{2}$. Then $d$ is given by:

$$d=\frac{2bc\cos{\frac{\alpha}{2}}}{b+c}\tag{2}$$

Proof. Let $BD=m$ and $CD=n$. We make use of the Angle Bisector Theorem and the Law of Cosines. Indeed, by the Angle Bisector Theorem we have

$$\frac{b}{c}=\frac{n}{m}$$

which can also be written as

$$\frac{b^2}{c^2}=\frac{n^2}{m^2}\tag{3}$$

Now, by the Law of Cosines, 

$$n^2=b^2+d^2-2bd\cos{\frac{\alpha}{2}}\tag{4}$$

$$m^2=c^2+d^2-2cd\cos{\frac{\alpha}{2}}\tag{5}$$

Dividing $(4)$ by $(5)$ and substituting $\frac{n^2}{m^2}$ by $\frac{b^2}{c^2}$ we have

$$\frac{b^2}{c^2}=\frac{b^2+d^2-2bd\cos{\frac{\alpha}{2}}}{c^2+d^2-2cd\cos{\frac{\alpha}{2}}}$$

From which we get

$$d^2(b^2-c^2)=2bcd\cos{\frac{\alpha}{2}}(b-c)$$

Isolating $d$, factoring and simplifying we obtain $(2)$.

$\square$

Another proof of Lemma 1 can be found here.

Lemma 2. Let $\triangle{ABC}$ be a triangle and let $a$, $b$ and $c$ be the lengths of sides $BC$, $AC$ and $BC$, respectively. Then the following identity holds

$$2\cos{\frac{\alpha}{2}}=\sqrt{\frac{(b+c)^2-a^2}{bc}}\tag{6}$$

A proof of Lemma 2 can be found here.

$\square$

Main proof

Combining $(2)$ and $(6)$ we obtain

$$d=\frac{bc}{(b+c)}\cdot{\sqrt{\frac{(b+c)^2-a^2}{bc}}}$$

Raising both sides to the power of 2 we get $(1)$.

$\square$

Related material

Length of Angle Bisector

Proving the length of angle bisector

domingo, 28 de marzo de 2021

Mis publicaciones

Este artículo inspiró la portada del CMJ, Volumen 53, 2022 - 2. Mire aquí.

En Forum Geometricorum

En MATINF

En Sangaku Journal of Mathematics

En Online Catalogue of Archimedean circles

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