Numerically Geometric

$|AX|:|XP|=1:1$

$|AX|:|AP|=1:2$

Let $|AX|=y$

Then, $|AP|=2y$

$|AQ|:|QB|=1:3$

$|AQ|:|AB|=1:4$

Let $|AQ|=x$

Then, $|AB|=4x$

$\Delta AQX \sim \Delta APB$

$\cfrac{|AQ|}{|AP|}=\cfrac{|AX|}{|AB|}$

$\cfrac{x}{2y}=\cfrac{y}{4x}$

$2y^{2}=4x^{2}$

$y^{2}=2x^{2}$

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In $\Delta AQX$ ,

$|QX|^{2}=y^{2}-x^{2}$

$|QX|^{2}=2x^{2}-x^{2}$

$|QX|^{2}=x^{2}$

$|QX|=x$

$|AX|=\sqrt{2}x$

$|XP|=\sqrt{2}x$

$|AP|=2\sqrt{2}x$

$\cfrac{Area of \Delta AQX}{Area of \Delta APB}=(\cfrac{|AQ|}{|AP|})^{2}=(\cfrac{1}{2\sqrt2})^{2}=\cfrac{1}{8}$

$\cfrac{Area of \Delta AQX}{Area of quadrilateral BQXP}=\cfrac{1}{7}$

$\cfrac{Area of quadrilateral BQXP}{Area of \Delta APB}=\cfrac{7}{8}$

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$\Delta AQX \sim \Delta XPC$

$\cfrac{Area of \Delta AQX}{Area of \Delta XPC}=(\cfrac{|AQ|}{|XP|})^{2}=(\cfrac{1}{\sqrt2})^{2}=\cfrac{1}{2}$

$\cfrac{Area of \Delta XPC}{Area of \Delta AXC}=\cfrac{|XP|}{|AX|}=\cfrac{1}{1}$

$\cfrac{Area of \Delta XPC}{Area of \Delta APC}=\cfrac{1}{2}$

$\cfrac{Area of \Delta AQX}{Area of \Delta APC}=\cfrac{1}{2}\times\cfrac{1}{2}=\cfrac{1}{4}$

$\cfrac{Area of \Delta APB}{Area of \Delta APC}=\cfrac{8}{4}$

$\cfrac{Area of quadrilateral BQXP}{Area of \Delta ABC}=\cfrac{7}{8+4}=\cfrac{7}{12}$

Area of $\Delta ABC=14\times \cfrac{12}{7}=24(units^{2})$