Let the points of intersections of the lines x - y + 1 = 0, x - 2y...

If $P, Q, R$ are mid-points of sides of $∆ ABC$ then $4 \times Area ( \Delta PQR ) = Area ( ∆ ABC )$

Area of \delta ABC = 4 \times \dfrac{1}{2}\left|\dfrac{{\left|\begin{cases}{a}_{1} & {b}_{1} & {c}_{1} \\ {a}_{2} & {b}_{2} & {c}_{2} \\ {a}_{3} & {b}_{3} & {c}_{3}\end{cases}\right|}^{2}}{{C}_{1}{C}_{2}{C}_{3}}\right|

Where ${C}_{1}, {C}_{2}, {C}_{3}$ are cofactors of${c}_{1}, {c}_{2}, {c}_{3}$

\delta ABC = 4 \times \dfrac{1}{2}\left|\dfrac{{\left|\begin{cases}1 & - 1 & 1 \\ 1 & - 2 & 3 \\ 2 & - 5 & 11\end{cases}\right|}^{2}}{( - 1 ) ( 3 ) ( - 1 )}\right|

$= 2 \times 9 \div 3$

$= 6$