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Take a Square II

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Richard of Mearns Castle High School sent us in this solution:
 

 Notice that $AE = \frac{1}{n} AB = \frac{1}{n}x$
 
We can say that $\triangle AEF$ and $\triangle DCF$ are similar. This is because $\angle AFE$ is equal to $\angle DFC$ (vertically opposite). Also $\angle FAE = \angle FCD$ and $\angle AEF = \angle CDF$ (alternate angles).
 
Since the triangles are similar we can say that the ratios of corresponding sides are the same. Therfore:
$$\eqalign{
\frac {DC}{AE} &= \frac{FC}{AF}\cr
\frac{x}{\frac{x}{n}}&= \frac{FC}{AF}\cr
n &= \frac{FC}{AF}
}$$
Hence$$FC = AF \times n $$
So $DE$ cuts $AC$ at the ratio $1:n$.