Problem 1

Let

ABCD

be a quadrilateral with all sides equal and

\angle ABC=60^\circ

. Let

\ell

be a line through

D

that does not meet the quadrilateral except at

D

. Let

E,F

be the intersections of

\ell

with

AB,BC

, respectively, and let

M=CE\cap AF

. Prove that

CA^2=CM\cdot CE

Step 2 of 5: Use the first similarity to obtain a length ratio

Previous step Next step

\triangle AED\sim\triangle CDF\Longrightarrow \frac{AE}{CD}=\frac{AD}{CF}\Longrightarrow \frac{AE}{AC}=\frac{AC}{CF}

Detailed analysis

The equilateral triangles give $AD\parallel BC$ and $CD\parallel AB$ . Since $E,D,F$ are collinear, triangles $AED$ and $CDF$ are similar. Thus $AE/CD=AD/CF$ , and replacing $AD$ and $CD$ by $AC$ gives $AE/AC=AC/CF$ .