Notăm d = distanța de la punctul D, situat pe bisectoare, la laturile
BA și BC ale unghiului B.
[tex]\it \mathcal{A}_{ABD}=\dfrac{cd}{2}\\ \\ \\ \mathcal{A}_{BDC}=\dfrac{ad}{2}\\ \\ \\ \mathcal{A}_{ABD}+\mathcal{A}_{BDC}=\mathcal{A}_{ABC} \Rightarrow\dfrac{cd}{2}+\dfrac{ad}{2}=\dfrac{bc}{2} \Rightarrow cd+ad=bc \Rightarrow d(a+c)=bc \Rightarrow \\ \\ \\ \Rightarrow d=\dfrac{bc}{a+c}[/tex]
[tex]\it \mathcal{A}_{ABD}=\dfrac{c\cdot\dfrac{bc}{a+c}}{2}=\dfrac{bc^2}{2(a+c)}\\ \\ \\ \mathcal{A}_{BDC}=\dfrac{a\cdot\dfrac{bc}{a+c}}{2}=\dfrac{abc}{2(a+c)}[/tex]
