area of qudirilateral

**X1337** · April 13th 2009, 09:17 AM

In triangle ABC, E is a point on AC and F is a point on AB. The lines BE and CF intersect at D. If the areas of triangles BDF, BCD nad CDE are 3,7 and 7 respectively, what is the area of the quadrilateral AEDF

**red_dog** · April 13th 2009, 12:20 PM

$area(BDC)=area(CDE)\Rightarrow BD=DE\Rightarrow area(ABD)=area(ADE)$

Let $area(ADE)=x, \ area(ADF)=y$ . Then, $x=y+3$

$\frac{area(BDF)}{area(BDC)}=\frac{FD}{DC}=\frac{3} {7}=\frac{area(AFD)}{area(ADC)}$

Then $\frac{y}{x+7}=\frac{3}{7}\Rightarrow 7y=3x+21$

Solving the system $\left\{\begin{array}{ll}x=y+3\\7y=3x+21\end{array} \right.$ we get $x=\frac{21}{2}, \ y=\frac{15}{2}$

$area(AEDF)=x+y=18$

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