O problema cu o bisectoare....

[math4]

Pb:In triunghiul $ABC$ avem $AB=36$ , $BC=40$ si $AC=44$.Bisectoarea $\angle A$ intersecteaza $BC$ in punctul $D$ si cercul circumscris triunghiului in $E$ (diferit de $A$).Calculeaza valoarea $DE^2$.

[mihai miculita]

1). $\widehat{BAD}\equiv\widehat{DAC}\Rightarrow\dfrac{|DB|}{|DC|}=\dfrac{|AB|}{|AC|}=\dfrac{9}{11}\Rightarrow\dfrac{|DB|}{|BC|}=\dfrac{|DB|}{|DB|+|DC|}=$$\dfrac{9}{9+11}=\dfrac{9}{20}\Rightarrow|DB|=\dfrac{9}{20}.|BC|=\dfrac{9}{20}.40=18\Rightarrow$
$\Rightarrow\boxed{|DB|=18}\Rightarrow\boxed{|DC|=|BC|-|DB|=40-18=22}.$
2). $|AD|^2=|AB|.|AC|-|DB|.|DC|=$$36.44-18.22=4.(18.22)-18.22=$$3.18.22=36.33$$\Rightarrow\boxed{|AD|=6\sqrt{33}}.$
3). $ABEC-inscriptibil\Rightarrow|DA|.|DE|=|DB|.|DC|\Rightarrow|DE|=$$\dfrac{|DB|.|DC|}{|DA|}=\dfrac{18.22}{6\sqrt{33}}=2\sqrt{33}\Rightarrow\boxed{|DE|=132}.$