I need to find the area of the shaded region in the diagram below. I used a geometry software with different measures for $AB$ and found out that there is a pattern for the shaded area, for example when $AB=10$ the area is $25$ and when $AB=20$ the area is $100$ and for $AB=30$ the area is $225$ and finally for $AB=40$ the area is $400$. But I don't really know how to calculate the shaded area.
Best Answer
Denote by $P$ projection of $D$ onto $CF$. We have by Pythagoras' theorem: $$CF \cdot (CP-PF)=(CP+PF)(CP-PF)=CP^2-PF^2=(CD^2-DP^2)-(FD^2-DP^2)=CD^2-FD^2=BC^2+BD^2-FD^2=BC^2$$ So we obtain the following system of equations: $$\begin{cases} CP+PF=CF \\ CP-PF=\frac{BC^2}{CF} \end{cases}$$ Subtracting second from the first and dividing by two we get that: $$PF=\frac{CF-\frac{BC^2}{CF}}{2}=\frac{CF^2-BC^2}{2CF}=\frac{CA^2-BC^2}{2EF}=\frac{AB^2}{2EF}$$ Now, since the triangles $DEF$ and $PEF$ have the same base and equal heights they have to have the same area, therefore we have: $$[DEF]=[PEF]=\frac{EF \cdot PF}{2}=\frac{AB^2}{4}$$ Plugging $AB=10$ we obtain $[DEF]=25$.