Points [tex]D, E, F[/tex] are on the sides [tex]BC, CA, AB[/tex] respectively of a triangle [tex]ABC[/tex] such that [tex]\overrightarrow{BD}=\lambda \overrightarrow{BC}, \overrightarrow{CE}=\lambda \overrightarrow{CA}, -\overrightarrow{FA}=\lambda \overrightarrow{AB}[/tex]. where [tex]\lambda[/tex] is a scalar. The lines [tex]BE[/tex] and [tex]CF[/tex] meet at [tex]P[/tex], the lines [tex]CF[/tex] and [tex]AD[/tex] meet at [tex]Q[/tex], and the lines [tex]AD[/tex] and [tex]BE[/tex] meet at [tex]R[/tex]. The position vectors of [tex]A, B, C[/tex] are [tex]\mathbf{a}, \mathbf{b}, \mathbf{c}[/tex] respectively, with respect to an origin outside the plane [tex]ABC[/tex].

(a) Show that the area [tex]\Delta[/tex] of triangle [tex]ABC[/tex] is given by [tex]\Delta=\frac{1}{2}|\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a} +\mathbf{a} \times \mathbf{b}|[/tex}.


(b) Show that the position vector of [tex]P[/tex] is [tex]\frac{\lambda(1-\lambda)\mathbf{a}+\lambda^2 \mathbf{b}+(1-\lambda)^2 \mathbf{c}}{1-\lambda +\lambda ^2}[/tex].


(c) Show that [tex]\overrightarrow{PQ}=\frac{1-2\lambda}{1-\lambda+\lambda^2} [(1-\lambda)\mathbf{a}+\lambda \mathbf{b}-\mathbf{c}][/tex].


(d) Show that the area of triangle [tex]PQR[/tex] is [tex]\frac{(1-2\lambda)^2}{1-\lambda +\lambda^2} \Delta[/tex].